Vectors in the plane practice problems
Component form, length, unit vectors, combinations, and the direction angle of plane vectors. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Find the unit vector in the direction of \( \displaystyle \mathbf v = \left\langle -2, -2 \right\rangle \).
Find \( \displaystyle \overrightarrow{PQ} \) in component form and its length, for \( \displaystyle P(5, 2) \) and \( \displaystyle Q(-3, 6) \).
Find the component form of the vector \( \displaystyle \mathbf u \) with \( \displaystyle \|\mathbf u\| = 2 \) at angle \( \displaystyle \theta = \frac{11 \pi}{6} \) from the positive \( \displaystyle x \)-axis.
Find the unit vector in the direction of \( \displaystyle \mathbf v = \left\langle -2, 2 \right\rangle \).
For \( \displaystyle \mathbf a = \left\langle -5, 5 \right\rangle \) and \( \displaystyle \mathbf b = \left\langle 2, 0 \right\rangle \), find \( \displaystyle -3\mathbf a + 3\mathbf b \) and its magnitude.
Find the angle \( \displaystyle \theta \in [0, 2\pi) \) that \( \displaystyle \mathbf u = \left\langle \sqrt{3}, 1 \right\rangle \) makes with the positive \( \displaystyle x \)-axis.
For \( \displaystyle \mathbf a = \left\langle -3, 6 \right\rangle \) and \( \displaystyle \mathbf b = \left\langle 5, 5 \right\rangle \), find \( \displaystyle -4\mathbf a + 1\mathbf b \) and its magnitude.
Find the angle \( \displaystyle \theta \in [0, 2\pi) \) that \( \displaystyle \mathbf u = \left\langle \frac{5}{2}, \frac{5 \sqrt{3}}{2} \right\rangle \) makes with the positive \( \displaystyle x \)-axis.
Find the unit vector in the direction of \( \displaystyle \mathbf v = \left\langle 4, -4 \right\rangle \).
Find the unit vector in the direction of \( \displaystyle \mathbf v = \left\langle -6, -8 \right\rangle \).