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5 problems match “unit vectors”. Clear

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1. Angle between a unit vector and a given vector ●○○○○ 10 ptsLet $\overrightarrow{a}$ be a unit vector and $\overrightarrow{b}=(1, \sqrt{3})$. Given that $\overrightarrow{a} \cdot \overrightarrow{b}=1$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ in degrees.
2. Cross product of two vectors ●○○○○ 10 ptsFind $(2\hat{i}+3\hat{j}-\hat{k})\times(3\hat{j}+2\hat{k})$. Write the result as $a\hat{i}+b\hat{j}+c\hat{k}$ and enter the coefficients $a$, $b$, $c$.
3. Area of parallelogram from its diagonals ●●○○○ 20 ptsLet $\mathbf{p}$ and $\mathbf{q}$ be two three-dimensional unit vectors such that the angle between them is $30^\circ.$ Find the area of the parallelogram whose diagonals correspond to $\mathbf{p} + 2 \mathbf{q}$ and $2 \mathbf{p} + \mathbf{q}.$ Give your answer as a decimal.
4. Angles of a unit vector with the axes ●●○○○ 20 ptsA unit vector $\vec a$ makes an angle $\frac{\pi}{3}$ with $\hat i$, an angle $\frac{\pi}{4}$ with $\hat j$, and an acute angle $\theta$ with $\hat k$. Find $\theta$ in degrees, and hence find the components of $\vec a$ along $\hat i$, $\hat j$, $\hat k$, in that order. Round the second component to two decimal places.
5. Dot product with a cross product of unit vectors ●●●○○ 50 ptsLet $\mathbf{u}$ and $\mathbf{v}$ be unit vectors, and let $\mathbf{w}$ be a vector such that $\mathbf{u} \times \mathbf{v} + \mathbf{u} = \mathbf{w}$ and $\mathbf{w} \times \mathbf{u} = \mathbf{v}$. Compute $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$.