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24 problems match “vectors”. Clear

Cosine of angle between vectors ●●○○○ 20 ptsFor certain vectors $\mathbf{p}$ and $\mathbf{q}$ , the vectors $3 \mathbf{p} + \mathbf{q}$ and $5 \mathbf{p} - 3 \mathbf{q}$ are orthogonal. Also, the vectors $2 \mathbf{p} + \mathbf{q}$ and $4 \mathbf{p} - 2 \mathbf{q}$ are orthogonal. If $\theta$ is the angle between $\mathbf{p}$ and $\mathbf{q}$ , find $\cos \theta$ . Give your answer as a decimal.
Cosine of angle with orthogonal vector sum ●●○○○ 20 ptsLet $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be three vectors with equal magnitudes, such that any two of them are orthogonal. If $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{a} + \mathbf{b} + \mathbf{c},$ find $\cos \theta.$ Round to two decimal places.
Area of a space quadrilateral ●●○○○ 20 ptsThe points $A = (3,-4,2)$ , $B = (5,-8,5)$ , $C = (4,-3,0)$ , and $D = (6,-7,3)$ in space form a flat quadrilateral. Find the area of this quadrilateral. Round to two decimal places.
Area of triangle from transformed vertices ●●○○○ 20 ptsThe triangle with vertices $\vec{A},$ $\vec{B},$ and $\vec{C}$ has area 12. Find the area of the triangle whose vertices are $-\vec{A} + \vec{B} + \vec{C},$ $\vec{A} - \vec{B} + \vec{C},$ and $\vec{A} + \vec{B} - \vec{C}.$
Dot product from norms and cross product ●●○○○ 20 ptsLet $\mathbf{a}$ and $\mathbf{b}$ be vectors such that $\|\mathbf{a}\| = 2,$ $\|\mathbf{b}\| = 5,$ and $\|\mathbf{a} \times \mathbf{b}\| = 8.$ Find $|\mathbf{a} \cdot \mathbf{b}|.$
Angle CBD in a tetrahedron ●●○○○ 20 ptsIn tetrahedron $ABCD$ , $BC=1$ , $BD=2$ , $\angle ABC=90^{\circ}$ , and $\overrightarrow{BC}\cdot\overrightarrow{DA}=-\sqrt{3}$ . Find $\angle CBD$ in degrees.
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.
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.
Area of a parallelogram from vector combinations ●●○○○ 20 ptsCalculate the area of the parallelogram constructed on the vectors $\vec{a}=3\vec{p}+2\vec{q}$ and $\vec{b}=2\vec{p}-\vec{q}$ , given that $|\vec{p}|=4$ , $|\vec{q}|=3$ and the angle between $\vec{p}$ and $\vec{q}$ is $\frac{3\pi}{4}$ . Round to two decimal places.
Coefficients of a vector in a basis ●●○○○ 20 ptsGiven vectors $\vec{a} = (1, -1)$ , $\vec{b} = (2, 1)$ and $\vec{c} = (-2, 1)$ , suppose $\vec{c} = x\vec{a} + y\vec{b}$ with $x, y \in \mathbb{R}$ . Find $x - y$ .