Question

Find the projection of u = <–6, –7> onto v = <1, 1> a. <-13/2,-13/2> b. <39,91/2> c. <-13/1764,-13/1764> d. <-2/13,-2/13>

Answer 1

a. <-13/2,-13/2>

Explanation:

The projection of a vector u onto another vector v is given by;

`proj_vu` =
`((u.v)/(|v|^2))v` ----------------(i)

Where;

u.v is the dot product of vectors u and v

|v| is the magnitude of vector v

Given:

u = <-6, -7>

v = <1, 1>

These can be re-written in unit vector notation as;

u = -6i -7j

v = i + j

Now;

Let's find the following

(i) u . v

u . v = (-6i - 7j) . (i + j)

u . v = (-6i) (1i) + (-7j)(1j) [Remember that, i.i = j.j = 1]

u . v = -6 -7 = -13

(ii) |v|

|v| =
`√((1)^2 + (1)^2)`

|v| =
`√(2)`

Substitute these values into equation (i) as follows;

`proj_vu` =
`[(-13)/((√(2)) ^2)][i + j]`

`proj_vu` =
`(-13)/(2) [i + j]`

This can be re-written as;

`proj_vu` =
`(-13)/(2)i + (-13)/(2)j`

`proj_vu` =
`<(-13)/(2), (-13)/(2)>`