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High School
Algebra
Pre-Algebra
Factors and multiples

sputavanomr

Answered question

2021-12-03

Find the angle berween the two vectors, rounded to the nearest tenth of a degree:

u = i + j + k, v = 3 s - 4 k

Answer & Explanation

May Dunn

Beginner2021-12-04Added 12 answers

Step 1
Given Data:
Two vectors:
$u = i + j + k = (1, 1, 1)$
$v = 3 i - 4 k = (3, 0, - 4)$
The dot product of two vectors is,
$u \cdot v = (1, 1, 1) \cdot (3, 0, - 4)$
$= (1) (3) + (1) (0) + (1) (- 4)$
$= 3 + 0 - 4$
$= 3 - 4$
$= - 1$
Step 2
The value of vector u is,
$∥ u ∥= \sqrt{(1)^{2} + (1)^{2} + (1)^{2}}$
$= \sqrt{1 + 1 + 1}$
$= \sqrt{3}$
The value of vector v is,
$∥ u ∥= \sqrt{(3)^{2} + (0)^{2} + (- 4)^{2}}$
$= \sqrt{9 + 0 + 16}$
$= \sqrt{25}$
$= 5$
Step 3
The angle between two vectors is,
$c o s θ = \frac{u \cdot v}{∥ u ∥∥ v ∥}$
Substitute the values in the above equation.
$\cos θ = \frac{- 1}{(\sqrt{3}) (5)}$
$θ = \cos^{- 1} (- 0.1155)$
$θ = {96.6}^{\circ}$
Thus, the angle between two vectors is ${96.6}^{\circ}$

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