13.04 Vectors in 3D
State the set of points \left(x, y, z\right) defined by the following equation:
Find the distance between each of the following points:
\left( - 2 , 1, - 3 \right) and \left(0, - 1 , 2\right)
\left(0, 0, 0\right) and \left(2, 5, 1\right)
\left(3, - 2 , - 2 \right) and \left(2, 3, 4\right).
Simplify:
Rewrite the following vectors in terms of the unit vectors \mathbf{i}, \mathbf{j} and \mathbf{k}.
Rewrite the following vectors as an ordered triple:
\overrightarrow{AB} where A \left(3, - 1 , - 7 \right) and B \left(4, 5, - 2 \right)
Rewrite each of the following vectors as a single vector:
Find the magnitude of the following vectors:
If 2 \left( x \mathbf{i} - 9 \mathbf{j} + 2 \mathbf{k}\right) - 3 \left( - 5\mathbf{ }i + y \mathbf{j} + 5 \mathbf{k}\right) - \frac{1}{2} \left( 4 \mathbf{i} - 6 \mathbf{j} + z \mathbf{k}\right) = 25 \mathbf{i} - 24 \mathbf{j} - 12 \mathbf{k}, find the values of x, y and z.
Find the position vector of \mathbf{v}=\overrightarrow{PQ} for each of the following. Give your answer in the form a \mathbf{i} + b \mathbf{j} + c \mathbf{k}.
P = \left( - 3 , 5, 0\right) and Q = \left(9, 6, - 4 \right)
P = \left( - 7 , - 2 , 3\right) and Q = \left(8, - 9 , 1\right)
The initial point of \overrightarrow{AB}=\left( - 13 , 5, - 9 \right) is A \left(7, - 2 , 8\right). Find the coordinates of the terminal point B?
The terminal point of\overrightarrow{AB}=\left(-1, 1, 2\right) is B \left(7, - 4 , 3\right). Find the coordinates of the initial point A.
Find the equation of a sphere with centre \left(5, 2, - 6 \right) and radius 2 units.