Question

Use the alternative form of the dot product to find u · v.

Answer 1

Answer:

Step-by-step explanation:

u = 45

v = 30

Angle between them, θ = 5π / 6 = 150°

(a) The formula for the dot product is given by

$\overrightarrow{u}.\overrightarrow{v}= u v Cos\theta$

$\overrightarrow{u}.\overrightarrow{v}= 45 \times 30 \times Cos150$

$\overrightarrow{u}.\overrightarrow{v}= -1169.13$

(b) $\overrightarrow{u}=Cos\frac{\pi }{3}\widehat{i}+Sin\frac{\pi }{3}\widehat{j}$

$\overrightarrow{u}=0.5\widehat{i}+0.866\widehat{j}$

$\overrightarrow{v}=Cos\frac{2\pi }{3}\widehat{i}+Sin\frac{2\pi }{3}\widehat{j}$

$\overrightarrow{v}=-0.707\widehat{i}+0.707\widehat{j}$

Let the angle between them is θ

The formula in terms of the dot product is given by

$Cos\theta =\frac{\overrightarrow{u}.\overrightarrow{v}}{u v}$

$u=\sqrt{0.5^{2}+0.866^{2}}=1$

$v=\sqrt{-0.707^{2}+0.707^{2}}=1$

$Cos\theta =\frac{0.5 \times (-0.707) + 0.866 \times 0.707)}{1\times 1}$

$Cos\theta=0.2587$

θ = 75°

(c) $\overrightarrow{u}=1\widehat{i}+4\widehat{j}+8\widehat{k}$

$u=\sqrt{1^{2}+4^{2}+8^{2}}=9$

$Cos\alpha =\frac{1}{9}$

$Cos\beta =\frac{4}{9}$

$Cos\gamma =\frac{8}{9}$