Q 20.

Question

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

$u = <1, 2>, v = <3, 5>$

Step-by-Step Solution

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Answer

The dot product is 13 and the angle is $4.439 °$ .

1Step 1. Given information.

The given pairs of vectors are:

$u = <1, 2>, v = <3, 5>$

2Step 2. Find the dot product.

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

$u = <1, 2>, v = <3, 5> u \cdot v = 1 (3) + 2 (5) = 3 + 10 = 13$

Therefore, the dot product of the given two vectors $u = <1, 2>, v = <3, 5>$ is 13.

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

The formula for the angle $θ$ between the two vectors is:

$\cos θ = \frac{u \cdot v}{||u|| ||v||}$

$u = <1, 2> a n d v = <3, 5> u \cdot v = 13 ||u|| = \sqrt{1^{2} + 2^{2}} = \sqrt{1 + 4} = \sqrt{5} ||v|| = \sqrt{3^{2} + 5^{2}} = \sqrt{9 + 25} = \sqrt{34}$

Then,

$\cos θ = \frac{u \cdot v}{||u|| ||v||} = \frac{13}{\sqrt{5} \sqrt{34}} = \frac{13}{\sqrt{170}} θ = \cos^{- 1} (\frac{13}{\sqrt{170}}) = 4.439 °$

Therefore, the angle is $4.439 °$ .

Q .18.

Q 21.

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