Vectors in three dimensions practice problems
Vectors in three dimensions: between points, combinations, length, and a vector of given length along a direction. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
For \( \displaystyle \mathbf a = \left\langle 0, 1, 4 \right\rangle \) and \( \displaystyle \mathbf b = \left\langle -3, 4, -6 \right\rangle \), find \( \displaystyle 1\mathbf a - (-4)\mathbf b \) and \( \displaystyle \|\mathbf a - \mathbf b\| \).
Express \( \displaystyle \overrightarrow{PQ} \) for \( \displaystyle P(-3, -4, 2) \) and \( \displaystyle Q(-6, -5, 4) \) in component form, and find the unit vector in its direction.
For \( \displaystyle \mathbf a = \left\langle 6, 3, 1 \right\rangle \) and \( \displaystyle \mathbf b = \left\langle -3, 3, -6 \right\rangle \), find \( \displaystyle -4\mathbf a - (-2)\mathbf b \) and \( \displaystyle \|\mathbf a - \mathbf b\| \).
Express \( \displaystyle \overrightarrow{PQ} \) for \( \displaystyle P(0, 5, 5) \) and \( \displaystyle Q(-4, 0, -3) \) in component form, and find the unit vector in its direction.
For \( \displaystyle \mathbf a = \left\langle -4, -3, -4 \right\rangle \) and \( \displaystyle \mathbf b = \left\langle 2, 6, 6 \right\rangle \), find \( \displaystyle -3\mathbf a - (1)\mathbf b \) and \( \displaystyle \|\mathbf a - \mathbf b\| \).
For \( \displaystyle \mathbf a = \left\langle 0, 0, 4 \right\rangle \) and \( \displaystyle \mathbf b = \left\langle 3, 3, -1 \right\rangle \), find \( \displaystyle -2\mathbf a - (-2)\mathbf b \) and \( \displaystyle \|\mathbf a - \mathbf b\| \).
Find the vector \( \displaystyle \mathbf u \) with \( \displaystyle \|\mathbf u\| = 10 \) pointing in the same direction as \( \displaystyle \mathbf v = \left\langle 3, 0, -4 \right\rangle \).
Find the vector \( \displaystyle \mathbf u \) with \( \displaystyle \|\mathbf u\| = 2 \) pointing in the same direction as \( \displaystyle \mathbf v = \left\langle -2, 6, -9 \right\rangle \).
Find the vector \( \displaystyle \mathbf u \) with \( \displaystyle \|\mathbf u\| = 7 \) pointing in the same direction as \( \displaystyle \mathbf v = \left\langle 1, 1, -1 \right\rangle \).
Find the vector \( \displaystyle \mathbf u \) with \( \displaystyle \|\mathbf u\| = 9 \) pointing in the same direction as \( \displaystyle \mathbf v = \left\langle 0, 0, -5 \right\rangle \).