Finding Perpendicular Vectors using the Dot Product

Oct 13, 2025 by pinkandtrash.com 122 views

=====================================================

Introduction

The dot product is a fundamental concept in linear algebra that allows us to find the angle between two vectors. In this article, we will explore how to use the dot product to find perpendicular vectors. We will start by defining what it means for two vectors to be perpendicular and then introduce the dot product formula. Finally, we will apply the dot product to find the value of $k$ in the given problem.

What are Perpendicular Vectors?

Two vectors are said to be perpendicular if their dot product is equal to zero. In other words, if we have two vectors $\mathbf{a}$ and $\mathbf{b}$ , then they are perpendicular if and only if $\mathbf{a} \cdot \mathbf{b} = 0$ . This is a crucial concept in linear algebra, as it allows us to find the angle between two vectors.

The Dot Product Formula

The dot product formula is given by:

\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta

where $\mathbf{a}$ and $\mathbf{b}$ are two vectors, $|\mathbf{a}|$ and $|\mathbf{b}|$ are their magnitudes, and $\theta$ is the angle between them.

Finding Perpendicular Vectors using the Dot Product

Now that we have introduced the dot product formula, we can use it to find perpendicular vectors. Let's consider two vectors $\mathbf{a} = \langle 9, 8 \rangle$ and $\mathbf{b} = \langle -3, k \rangle$ . We are given that these vectors are perpendicular, so we can set up the equation:

\mathbf{a} \cdot \mathbf{b} = 0

Using the dot product formula, we can expand this equation as follows:

\langle 9, 8 \rangle \cdot \langle -3, k \rangle = 0

9(-3) + 8k = 0

-27 + 8k = 0

Solving for $k$

Now that we have set up the equation, we can solve for $k$ . To do this, we can add 27 to both sides of the equation:

8k = 27

Next, we can divide both sides of the equation by 8:

k = \frac{27}{8}

Therefore, the value of $k$ is $\boxed{\frac{27}{8}}$ .

Conclusion

In this article, we have used the dot product to find perpendicular vectors. We started by defining what it means for two vectors to be perpendicular and then introduced the dot product formula. Finally, we applied the dot product to find the value of $k$ in the given problem. We hope that this article has provided a clear and concise explanation of how to use the dot product to find perpendicular vectors.

Applications of the Dot Product

The dot product has many applications in mathematics and physics. Some of these applications include:

Finding the angle between two vectors
Finding the magnitude of a vector
Finding the projection of one vector onto another
Finding the work done by a force on an object

Example Problems

Here are a few example problems that involve finding perpendicular vectors using the dot product:

Find the value of $k$ in the equation $\langle 2, 3 \rangle \cdot \langle -4, k \rangle = 0$ .
Find the value of $x$ in the equation $\langle 1, 2 \rangle \cdot \langle x, 3 \rangle = 0$ .
Find the value of $y$ in the equation $\langle 3, 4 \rangle \cdot \langle 2, y \rangle = 0$ .

Exercises

Here are a few exercises that involve finding perpendicular vectors using the dot product:

Find the value of $k$ in the equation $\langle 5, 6 \rangle \cdot \langle -2, k \rangle = 0$ .
Find the value of $x$ in the equation $\langle 3, 4 \rangle \cdot \langle x, 2 \rangle = 0$ .
Find the value of $y$ in the equation $\langle 2, 3 \rangle \cdot \langle 4, y \rangle = 0$ .

Solutions

Here are the solutions to the exercises:

Find the value of $k$ in the equation $\langle 5, 6 \rangle \cdot \langle -2, k \rangle = 0$ .

\langle 5, 6 \rangle \cdot \langle -2, k \rangle = 0

5(-2) + 6k = 0

-10 + 6k = 0

6k = 10

k = \frac{10}{6}

k = \frac{5}{3}

Find the value of $x$ in the equation $\langle 3, 4 \rangle \cdot \langle x, 2 \rangle = 0$ .

\langle 3, 4 \rangle \cdot \langle x, 2 \rangle = 0

3x + 4(2) = 0

3x + 8 = 0

3x = -8

x = -\frac{8}{3}

Find the value of $y$ in the equation $\langle 2, 3 \rangle \cdot \langle 4, y \rangle = 0$ .

\langle 2, 3 \rangle \cdot \langle 4, y \rangle = 0

2(4) + 3y = 0

8 + 3y = 0

3y = -8

y = -\frac{8}{3}$$<br/> # Q&A: Using the Dot Product to Find Perpendicular Vectors =====================================================

Frequently Asked Questions

Q: What is the dot product?

A: The dot product is a mathematical operation that takes two vectors as input and produces a scalar value as output. It is used to find the angle between two vectors, as well as to find the magnitude of a vector.

Q: How do I find the dot product of two vectors?

A: To find the dot product of two vectors, you can use the formula:

\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta </span></p> <p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">a</mi></mrow><annotation encoding="application/x-tex">\mathbf{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">a</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">b</mi></mrow><annotation encoding="application/x-tex">\mathbf{b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbf">b</span></span></span></span> are the two vectors, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="bold">a</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|\mathbf{a}|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathbf">a</span><span class="mord">∣</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="bold">b</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|\mathbf{b}|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathbf">b</span><span class="mord">∣</span></span></span></span> are their magnitudes, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> is the angle between them.</p> <h3>Q: What is the significance of the dot product?</h3> <hr> <p>A: The dot product has many significant applications in mathematics and physics. Some of these applications include:</p> <ul> <li>Finding the angle between two vectors</li> <li>Finding the magnitude of a vector</li> <li>Finding the projection of one vector onto another</li> <li>Finding the work done by a force on an object</li> </ul> <h3>Q: How do I find the value of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> in the equation <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">b</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathbf{a} \cdot \mathbf{b} = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord mathbf">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbf">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>?</h3> <hr> <p>A: To find the value of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> in the equation <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">b</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathbf{a} \cdot \mathbf{b} = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord mathbf">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbf">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>, you can use the formula:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">b</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mi mathvariant="bold">a</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi mathvariant="bold">b</mi><mi mathvariant="normal">∣</mi><mi>cos</mi><mo>⁡</mo><mi>θ</mi></mrow><annotation encoding="application/x-tex">\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord mathbf">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbf">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathbf">a</span><span class="mord">∣∣</span><span class="mord mathbf">b</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span></span></p> <p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">a</mi></mrow><annotation encoding="application/x-tex">\mathbf{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">a</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">b</mi></mrow><annotation encoding="application/x-tex">\mathbf{b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbf">b</span></span></span></span> are the two vectors, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="bold">a</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|\mathbf{a}|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathbf">a</span><span class="mord">∣</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="bold">b</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|\mathbf{b}|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathbf">b</span><span class="mord">∣</span></span></span></span> are their magnitudes, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> is the angle between them.</p> <h3>Q: What is the relationship between the dot product and the angle between two vectors?</h3> <hr> <p>A: The dot product is directly proportional to the cosine of the angle between two vectors. In other words, the dot product is equal to the product of the magnitudes of the two vectors and the cosine of the angle between them.</p> <h3>Q: Can the dot product be used to find the magnitude of a vector?</h3> <hr> <p>A: Yes, the dot product can be used to find the magnitude of a vector. If we have a vector <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">a</mi></mrow><annotation encoding="application/x-tex">\mathbf{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">a</span></span></span></span> and we know its dot product with another vector <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">b</mi></mrow><annotation encoding="application/x-tex">\mathbf{b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbf">b</span></span></span></span>, we can use the formula:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="bold">a</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mfrac><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">b</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="bold">b</mi><mi mathvariant="normal">∣</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">|\mathbf{a}| = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathbf">a</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3074em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathbf">b</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathbf">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <p>to find the magnitude of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">a</mi></mrow><annotation encoding="application/x-tex">\mathbf{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">a</span></span></span></span>.</p> <h3>Q: Can the dot product be used to find the projection of one vector onto another?</h3> <hr> <p>A: Yes, the dot product can be used to find the projection of one vector onto another. If we have two vectors <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">a</mi></mrow><annotation encoding="application/x-tex">\mathbf{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">a</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">b</mi></mrow><annotation encoding="application/x-tex">\mathbf{b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbf">b</span></span></span></span>, we can use the formula:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mtext>proj</mtext><mi mathvariant="bold">b</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">a</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">b</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="bold">b</mi><msup><mi mathvariant="normal">∣</mi><mn>2</mn></msup></mrow></mfrac><mi mathvariant="bold">b</mi></mrow><annotation encoding="application/x-tex">\text{proj}_{\mathbf{b}}(\mathbf{a}) = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|^2} \mathbf{b} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord text"><span class="mord">proj</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.242em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">b</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">a</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3074em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathbf">b</span><span class="mord"><span class="mord">∣</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathbf">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathbf">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathbf">b</span></span></span></span></span></p> <p>to find the projection of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">a</mi></mrow><annotation encoding="application/x-tex">\mathbf{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">a</span></span></span></span> onto <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">b</mi></mrow><annotation encoding="application/x-tex">\mathbf{b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbf">b</span></span></span></span>.</p> <h3>Q: Can the dot product be used to find the work done by a force on an object?</h3> <hr> <p>A: Yes, the dot product can be used to find the work done by a force on an object. If we have a force <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">F</mi></mrow><annotation encoding="application/x-tex">\mathbf{F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">F</span></span></span></span> and an object that is moving with a velocity <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">v</mi></mrow><annotation encoding="application/x-tex">\mathbf{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">v</span></span></span></span>, we can use the formula:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>W</mi><mo>=</mo><mi mathvariant="bold">F</mi><mo>⋅</mo><mi mathvariant="bold">v</mi></mrow><annotation encoding="application/x-tex">W = \mathbf{F} \cdot \mathbf{v} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">F</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">v</span></span></span></span></span></p> <p>to find the work done by the force on the object.</p> <h2>Additional Resources</h2> <hr> <ul> <li><a href="https://en.wikipedia.org/wiki/Dot_product">Wikipedia: Dot product</a></li> <li><a href="https://www.khanacademy.org/math/linear-algebra/linear-algebra-tutorial/dot-product/v/dot-product">Khan Academy: Dot product</a></li> <li><a href="https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/">MIT OpenCourseWare: Linear Algebra</a></li> </ul> <h2>Conclusion</h2> <hr> <p>In this article, we have answered some of the most frequently asked questions about using the dot product to find perpendicular vectors. We have covered topics such as the dot product formula, the significance of the dot product, and how to find the value of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> in the equation <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">b</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathbf{a} \cdot \mathbf{b} = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord mathbf">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathbf">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>. We hope that this article has provided a clear and concise explanation of the dot product and its applications.</p>