Question

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

Answer 1

Answer:

The angle between them is 60 degrees

Step-by-step explanation:

Given

$a = 2i + j -3k$

$b = 3i - 2j -k$

Required

The angle between them

The cosine of the angle between them is:

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

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|

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

$|a| = \sqrt{14$

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

$|b| = \sqrt{14}$

Recall that:

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

This gives:

$\cos(\theta) = \frac{7}{\sqrt{14} * \sqrt{14}}$

$\cos(\theta) = \frac{7}{14}$

$\cos(\theta) = 0.5$

Take arccos of both sides

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

$\theta =60^o$