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24 problems match “vectors”. Clear

Maximum angle between cyclic vectors ●●●○○ 50 ptsLet $a, b, c$ be nonzero real numbers. Find the maximum angle, in degrees, between the vectors $\begin{pmatrix} a \\ b \\ c \end{pmatrix}$ and $\begin{pmatrix} b \\ c \\ a \end{pmatrix}$ .
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 $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$ .
Sine of angle from triple product ●●●○○ 50 ptsLet $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be nonzero vectors, no two of which are parallel, such that $(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = \frac{1}{3} \|\mathbf{b}\| \|\mathbf{c}\| \mathbf{a}.$ Let $\theta$ be the angle between $\mathbf{b}$ and $\mathbf{c}.$ Find $\sin \theta.$ Round to two decimal places.
Coefficient from line-plane angle ●●●○○ 50 ptsThe angle between the plane $3x + by - 5z = 2$ and the straight line $x + 2 = \frac{3 - y}{2} = \frac{z - 1}{2}$ is $45^\circ$ . Find the constant $b$ , given that $0 < b < 10$ .