Vectors in the plane and in space

Given $\vec{u}=(2,-2,3)$ and $\vec{v}=(-3,-1,3)$, compute the vector $3\vec{u}-2\vec{v}$ and its magnitude.

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Given the points $A=(3,-1,2)$ and $B=(-1,1,6)$, find the distance $d(A,B)$, the midpoint of the segment $AB$ and the unit vector of $\overrightarrow{AB}$.

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Write the vector $\vec{w}=(8,1)$ as a linear combination of the vectors $\vec{u}=(2,1)$ and $\vec{v}=(1,-1)$.

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Compute the angle between the vectors $\vec{u}=(0,1,-1)$ and $\vec{v}=(1,-2,1)$.

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Compute the angles of the triangle with vertices $A=(1,2,0)$, $B=(2,2,1)$, $C=(1,3,1)$.

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Find the values of $k\in \mathbb{R}$ for which the vectors $\vec{u}=(k,1,k)$ and $\vec{v}=(k,-8,2)$ are orthogonal.

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Decompose $\vec{v}=(0,4,5)$ as the sum of a vector parallel to $\vec{u}=(1,2,2)$ and a vector orthogonal to $\vec{u}$.

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Find the unit vectors orthogonal to both $\vec{u}=(2,-2,-1)$ and $\vec{v}=(1,-2,0)$.

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Compute the area of the triangle with vertices $A=(1,-1,2)$, $B=(0,1,0)$, $C=(3,3,2)$.

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Compute the distance of the origin $O=(0,0,0)$ from the line through $A=(2,1,0)$ and $B=(0,-1,-1)$.

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Find the values of $k\in \mathbb{R}$ for which the vectors $\vec{u}=(k,2,-1)$ and $\vec{v}=(8,k,-2)$ are parallel.

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Compute the volume of the tetrahedron with vertices $A=(1,1,1)$, $B=(2,1,3)$, $C=(1,3,2)$, $D=(3,2,1)$.

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Find the values of $k\in \mathbb{R}$ for which the vectors $\vec{u}=(1,1,k)$, $\vec{v}=(1,k,1)$, $\vec{w}=(k,1,1)$ are coplanar.

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