Chapter 9: Q. 54 (page 622)

In Problems 51–58, find the dot product v.w and the angle between v and w.
$v = 2 i + 2 j - k; w = i + 2 j + 3 k$

Short Answer

Expert verified

The dot product of the vectors

$v = 2 i + 2 j - k; w = i + 2 j + 3 k$

is 3 and the angle between them is $74 . 5^{\circ}$ .

Step by step solution

Step 1. Given

The vectors:

$v = 2 i + 2 j - k; w = i + 2 j + 3 k$

To find the dot product and the angle between them.

Step 2. Dot product of the vectors.

The dot product is given by,

$v . w = (2 i + 2 j - k) . (i + 2 j + 3 k) = (2) (1) + (2) (2) + (- 1) (3) = 3$

Step 3. Angle between the vectors.

Ifu andv are two nonzero vectors, the angle $θ, 0 \leq θ \leq π$ betweenu andv is determined by the formula, $\cos θ = \frac{u . v}{||u|| ||v||}$

Step 4. Find the length of the vectors.

Now,

$||v|| = ||2 i + 2 j - k|| = \sqrt{2^{2} + 2^{2} + {(- 1)}^{2}} = \sqrt{9} = 3$
$||w|| = ||i + 2 j + 3 k|| = \sqrt{1^{2} + 2^{2} + 3^{2}} = \sqrt{14}$

Step 5. find the angle between given vectors

Angle between the vectors is given by,

$\cos θ = \frac{v . w}{||v|| ||w||} = \frac{3}{\sqrt{9} \sqrt{14}} = \frac{3}{3 \sqrt{14}} θ = \cos^{- 1} (\frac{1}{\sqrt{14}}) ≃ 74.5 °$

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In Problems 51–58, find the dot product v.w and the angle between v and w.
$v = 2 i + 2 j - k; w = i + 2 j + 3 k$

Short Answer

Step by step solution

Step 1. Given

Step 2. Dot product of the vectors.

Step 3. Angle between the vectors.

Step 4. Find the length of the vectors.

Step 5. find the angle between given vectors

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In Problems 51–58, find the dot product v.w and the angle between v and w.v=2i+2j-k;w=i+2j+3k

Short Answer

Step by step solution

Step 1. Given

Step 2. Dot product of the vectors.

Step 3. Angle between the vectors.

Step 4. Find the length of the vectors.

Step 5. find the angle between given vectors

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Discrete Mathematics

Mechanics Maths

Pure Maths

Calculus

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 51–58, find the dot product v.w and the angle between v and w.
$v = 2 i + 2 j - k; w = i + 2 j + 3 k$