Find the angle between the vectors ????=????+???? and ????=−????+????. (Give an exact answer. Use symbolic notation and fractions where needed.)

Question

asked Aug 14, 2022 6.6k views

1 Answer

Samuel Marks · Answer 1 · 2022-08-20T08:34:46+0000

Answer:

The angle between them is 60 degrees

Explanation:

Given

a = 2i + j -3k

b = 3i - 2j -k

Required

The angle between them

The cosine of the angle between them is:

$\cos(\theta) = (a\cdot b)/(|a|\cdot |b|)$

First, calculate a.b

$a \cdot b =(2i + j -3k) \cdot (3i - 2j -k)$

Multiply the coefficients of like terms

$a \cdot b =2 * 3 - 1 * 2 - 3 * -1$

$a \cdot b =7$

Next, calculate |a| and |b|

$|a| = \sqrt{2^2 + 1^2 + (-3)^2$

$|a| = \sqrt{14$

|b| = √(3^2 + (-2)^2 + (-1)^2)

|b| = √(14)

Recall that:

$\cos(\theta) = (a\cdot b)/(|a|\cdot |b|)$

This gives:

$\cos(\theta) = (7)/(√(14) * √(14))$

$\cos(\theta) = (7)/(14)$

$\cos(\theta) = 0.5$

Take arccos of both sides

$\theta =\cos^(-1)(0.5)$

$\theta =60^o$