MAGNITUDE OF A CARTESIAN VECTOR

Example 1

Find the magnitude of the vector x = 4i - 2j + 3k, the unit vector in the direction of x and the vector of magnitude 5 in the direction opposite to x.

Solution

We have |x| =  =  = .

Therefore the unit vector in the direction of x is the vector

and the vector of magnitude 5 in the direction opposite to x is

Example 2

Given the vectors u = 3i - j + 2k and v = 2i + 3j - k, find Cartesian forms of the vectors u + v, u - v, 2u + 3v.

Solution

First,

Next,

and finally,

Example 3

Given that P₁(-1, 2, 3) and P₂(3, 3, 4) are two points in space, find the Cartesian form of the vector .

Solution

Example 4

Show that the vectors

and

have the same magnitude.

Solution

Both vectors are in Cartesian form and their lengths can be calculated using the formula

We have

and

Therefore two given vectors have the same length.