The Vectors G = ( 5 B + 8 ) G=\begin{pmatrix}5 \\ B+8\end{pmatrix} G = ( 5 B + 8 ​ ) And H = ( 3 − B ) H=\begin{pmatrix}3 \\ -b\end{pmatrix} H = ( 3 − B ​ ) Are Perpendicular. Find The Value Of B B B .

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Introduction

In this article, we will explore the concept of perpendicular vectors and how to find the value of a variable in a given vector equation. We will use the dot product of two vectors to determine the value of $b$ in the given vectors $g$ and $h$. The dot product is a fundamental concept in linear algebra and is used to find the angle between two vectors.

What are Perpendicular Vectors?

Perpendicular vectors are vectors that have an angle of 90 degrees between them. In other words, if two vectors are perpendicular, their dot product is equal to zero. The dot product of two vectors $a = \begin{pmatrix}a_1 \\ a_2\end{pmatrix}$ and $b = \begin{pmatrix}b_1 \\ b_2\end{pmatrix}$ is given by the formula:

$$a \cdot b = a_1b_1 + a_2b_2$$

The Dot Product of Two Vectors

The dot product of two vectors is a scalar value that represents the amount of "similarity" between the two vectors. If the dot product is positive, it means that the vectors are pointing in the same direction. If the dot product is negative, it means that the vectors are pointing in opposite directions. If the dot product is zero, it means that the vectors are perpendicular.

Finding the Value of $b$

We are given two vectors $g=\begin{pmatrix}5 \\ b+8\end{pmatrix}$ and $h=\begin{pmatrix}3 \\ -b\end{pmatrix}$ that are perpendicular. To find the value of $b$, we can use the dot product formula:

$$g \cdot h = 5(3) + (b+8)(-b)$$

Since the vectors are perpendicular, their dot product is equal to zero:

$$5(3) + (b+8)(-b) = 0$$

Simplifying the equation, we get:

$$15 - b^2 - 8b = 0$$

Rearranging the equation, we get:

$$b^2 + 8b - 15 = 0$$

Solving the Quadratic Equation

We can solve the quadratic equation using the quadratic formula:

$$b = \frac{-8 \pm \sqrt{8^2 - 4(1)(-15)}}{2(1)}$$

Simplifying the equation, we get:

$$b = \frac{-8 \pm \sqrt{64 + 60}}{2}$$

$$b = \frac{-8 \pm \sqrt{124}}{2}$$

$$b = \frac{-8 \pm 2\sqrt{31}}{2}$$

$$b = -4 \pm \sqrt{31}$$

Conclusion

In this article, we used the dot product of two vectors to find the value of $b$ in the given vectors $g$ and $h$. We solved the quadratic equation using the quadratic formula and found two possible values for $b$. The value of $b$ is $-4 \pm \sqrt{31}$.

Final Answer

The final answer is $\boxed{-4 \pm \sqrt{31}}$.

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Introduction

In our previous article, we explored the concept of perpendicular vectors and how to find the value of a variable in a given vector equation. We used the dot product of two vectors to determine the value of $b$ in the given vectors $g$ and $h$. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is the dot product of two vectors?

A: The dot product of two vectors $a = \begin{pmatrix}a_1 \\ a_2\end{pmatrix}$ and $b = \begin{pmatrix}b_1 \\ b_2\end{pmatrix}$ is given by the formula:

$$a \cdot b = a_1b_1 + a_2b_2$$

Q: What is the significance of the dot product?

A: The dot product of two vectors is a scalar value that represents the amount of "similarity" between the two vectors. If the dot product is positive, it means that the vectors are pointing in the same direction. If the dot product is negative, it means that the vectors are pointing in opposite directions. If the dot product is zero, it means that the vectors are perpendicular.

Q: How do we find the value of $b$ in the given vectors $g$ and $h$?

A: We can use the dot product formula to find the value of $b$. Since the vectors are perpendicular, their dot product is equal to zero:

$$g \cdot h = 5(3) + (b+8)(-b) = 0$$

Simplifying the equation, we get:

$$15 - b^2 - 8b = 0$$

Rearranging the equation, we get:

$$b^2 + 8b - 15 = 0$$

We can solve the quadratic equation using the quadratic formula:

$$b = \frac{-8 \pm \sqrt{8^2 - 4(1)(-15)}}{2(1)}$$

Simplifying the equation, we get:

$$b = \frac{-8 \pm \sqrt{64 + 60}}{2}$$

$$b = \frac{-8 \pm \sqrt{124}}{2}$$

$$b = \frac{-8 \pm 2\sqrt{31}}{2}$$

$$b = -4 \pm \sqrt{31}$$

Q: What is the final answer for the value of $b$?

A: The final answer for the value of $b$ is $\boxed{-4 \pm \sqrt{31}}$.

Q: Can we use the dot product to find the angle between two vectors?

A: Yes, we can use the dot product to find the angle between two vectors. The formula for finding the angle between two vectors $a$ and $b$ is:

$$\cos(\theta) = \frac{a \cdot b}{|a||b|}$$

Where $\theta$ is the angle between the two vectors, and $|a|$ and $|b|$ are the magnitudes of the two vectors.

Q: What is the significance of the angle between two vectors?

A: The angle between two vectors represents the amount of "similarity" between the two vectors. If the angle is 0 degrees, it means that the vectors are pointing in the same direction. If the angle is 180 degrees, it means that the vectors are pointing in opposite directions. If the angle is 90 degrees, it means that the vectors are perpendicular.

Conclusion

In this article, we answered some frequently asked questions related to the topic of perpendicular vectors and the dot product. We hope that this article has provided you with a better understanding of the concept and how to apply it to solve problems.

Final Answer

The final answer is $\boxed{-4 \pm \sqrt{31}}$.