Exercises
- If you start at the initial point \((3,7,1)\) and travel to the terminal point \((1,2,5),\) along what vector did you travel?
- Let \(A\) be the point \((2,-9,1)\) and let \(B\) be the point \((3,2,-4)\) What are the components of the vector \(\overrightarrow{AB}?\) What are the components of the vector \(\overrightarrow{BA}?\)
- If you start at the initial point \((-2,7,6)\) and travel along a vector \(\langle -1,3,0 \rangle,\) what are the coordinates of the terminal point?
- Given points \(A\) and \(B\) such that the coordinates of \(A\) are \((0,-3,1)\) and such that \(\overrightarrow{AB} = \bigl\langle 10,2,-5 \bigr\rangle\) what must the coordinates of \(B\) be?
- Let \(\bm{v} = \langle 3,2,-4 \rangle.\) What is the magnitude of \(\bm{v}?\) What is \(\bm{\hat{v}},\) the unit vector in the direction of \(\bm{v}?\)
- What angle does \(\bm{v} = \langle 3,0,2 \rangle\) make with the \(z\)-axis?
- What angle does \(\bm{u} = \langle 1,3,-2 \rangle\) make with the \(xy\)-plane?
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Sketch the vectors \(\bm{u} = \langle 3,-3,5 \rangle\)
and \(\bm{v} = \langle 2,5,-4 \rangle\) in \(xyz\)-space,
then explicitly calculate each of the following:
\( \bm{u} + \bm{v} \)\( \bm{u} - \bm{v} \)\( |\bm{u}| \)\( |\bm{v}| \)\( \bm{\hat{u}} \)\( \bm{\hat{v}} \)\( 3\bm{u} \)\( \bm{u}\cdot\bm{v} \)\( \bm{u} - 2\bm{v} \)\( \bm{v} + \langle 1,-2,3 \rangle \)\( \bm{u} + 2\mathbf{j}-\mathbf{k} \)
- What is the measure of angle between \(\bm{u} = \langle 7,-2,1\rangle\) and \(\bm{v} = \langle 3,5,-3 \rangle?\) What is the area of the triangle framed by those vectors?
- What is the area of the triangle in space with vertices located at the coordinates \((1,1,1)\) and \((3,-3,8)\) and \((2,7,0)?\)