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

Answer:

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

Step-by-step explanation:

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

$proj_vu$ = $(\frac{u.v}{|v|^2})v$               ----------------(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| = $\sqrt{(1)^2 + (1)^2}$

|v| = $\sqrt{2}$

Substitute these values into equation (i) as follows;

$proj_vu$ = $[\frac{-13}{(\sqrt{2}) ^2}][i + j]$

$proj_vu$ = $\frac{-13}{2} [i + j]$

This can be re-written as;

$proj_vu$ = $\frac{-13}{2}i + \frac{-13}{2}j$

$proj_vu$ =