Refresher - Vectors in 3D Space

Vectors in 3-D Space

Example:

The vector OP has initial point at the origin O (0, 0, 0) and terminal point at P (2, 3, 5). We can draw the vectorOP as follows:

Magnitude of a 3-Dimensional Vector

Recall that the distance between 2 points in 3-dimensional space is

distance $A B = \sqrt{{(x_{s} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} + {(z_{2} - z_{1})}^{2}}$

Notation

The magnitude of a two dimensional vector $a = (a_{1}, a_{2})$ , denoted by $| | a | |$ (or sometimes by $| a |$ ), is given by
$| | a | | = | | (a_{1}, a_{2}) | | = \sqrt{{(a_{1})}^{2} + {(a_{2})}^{2}}$ .

For the vector OP above, the magnitude of the vector is given by:

$| OP | = \sqrt{(2^{2} + 3^{2} + 5^{2} = \sqrt{38})} = 6.16 units$

Angle Between 3-Dimensional Vectors

Earlier, we saw how to find the angle between 2-dimensional vectors. We use the same formula for 3-dimensional vectors:

$θ = a r c \frac{cos (p • Q)}{| P | | Q |}$

In the diagram on the right,

$α$ is the angle between u and the x-axis (in dark red),
$β$ is the angle between u and the y-axis (in green) and
$γ$ is the angle between u and the z-axis (in pink),

Notation

$i, j$ Form For Vectors

$a = (a_{1}, a_{2}) = a_{1} i + a_{2} j$