Find the angle between v and w. Round to one decimal place, if necessary.
-$\mathbf { v } = \mathrm { i } + \mathrm { j }$ and $\mathbf { w } = \mathrm { i } + \mathbf { j } - \mathbf { k }$
A) $66 ^ { \circ }$
B) $0 ^ { \circ }$
C) $35.3 ^ { \circ }$
D) $90 ^ { \circ }$

Question

Quizplus · Accepted Answer

The angle $\theta$ between two vectors $\mathbf{v}$ and $\mathbf{w}$ can be found using the dot product formula: $\mathbf{v} \cdot \mathbf{w} = \|\mathbf{v}\| \|\mathbf{w}\| \cos(\theta)$. For $\mathbf{v} = \mathrm{i} + \mathrm{j}$ and $\mathbf{w} = \mathrm{i} + \mathrm{j} - \mathrm{k}$, the dot product $\mathbf{v} \cdot \mathbf{w} = 1*1 + 1*1 + 0*(-1) = 2$. The magnitudes are $\|\mathbf{v}\| = \sqrt{1^2 + 1^2} = \sqrt{2}$ and $\|\mathbf{w}\| = \sqrt{1^2 + 1^2 + (-1)^2} = \sqrt{3}$. Thus, $\cos(\theta) = \frac{2}{\sqrt{2} \sqrt{3}} = \frac{\sqrt{6}}{6}$. Solving for $\theta$ gives $\theta \approx 35.3^\circ$.

Find the Angle Between V and W v=i+j\mathbf { v } = \mathrm { i } + \mathrm { j }v=i+j

Find the Angle Between V and W $\mathbf { v } = \mathrm { i } + \mathrm { j }$