Q 22.

Question

In Exercises 20-23, find the dot product of the given pairs of vectors and the angle between the two vectors.

$u = <3, - 1, 2>, v = <- 4, - 6, 3>$

Step-by-Step Solution

Verified

Answer

The dot product is 0 and the angle is $90 °$ .

1Step 1. Given information.

The given pairs of vectors are:

$u = <3, - 1, 2>, v = <- 4, - 6, 3>$

2Step 2. Find the dot product.

The dot product is: $u \cdot v = u_{1} v_{1} + u_{2} v_{2} + u_{3} v_{3}$

$u = <3, - 1, 2> and v = <- 4, - 6, 3> u \cdot v = 3 (- 4) + (- 1) (- 6) + 2 (3) = - 12 + 6 + 6 = 0$

Therefore, the dot product of the given two vectors $u = <3, - 1, 2> and v = <- 4, - 6, 3>$ is 0.

3Step 3. Find the angle between the two vectors.

The formula for the angle between the two vectors is:

$\cos θ = \frac{u . v}{||u|| ||v||} u = <3, - 1, 2> and v = <- 4, - 6, 3> u \cdot v = 0 ||u|| = \sqrt{3^{2} + {(- 1)}^{2} + 2^{2}} \sqrt{9 + 1 + 4} = \sqrt{14} ||v|| = \sqrt{{(- 4)}^{2} + {(- 6)}^{2} + 3^{2}} \sqrt{16 + 36 + 9} = \sqrt{61}$

Then,

$\cos θ = \frac{u . v}{||u|| ||v||} \cos θ = \frac{0}{\sqrt{14} \sqrt{61}} \cos θ = 0 θ = \cos^{- 1} (0) = 90 °$

Therefore, the angle is $90 °$ .

Q 21.

Q 23.

Other exercises in this chapter

Q 20.

In Exercises 20-23, find the dot product of the given pairs of vectors and the angle between the two vectors.u=1,2, v=3,5

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Q 21.

In Exercises 20-23, find the dot product of the given pairs of vectors and the angle between the two vectors.u=2,0,-5, v=-3,7,-1

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Q 23.

In Exercises 20-23, find the dot product of the given pairs of vectors and the angle between the two vectors.u=-5,1,3, v=-3,2,7

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Q 24.

In Exercises 24-27, find compu v, proju v, and the component of v orthogonal to u.u=1,2, v=3,5

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