For each of the following problems, sketch the vectors involved on a coordinate plane, then decompose vector into two vectors →v₁ and →v₂, where →v₁ is parallel to →w, and →v₂ is orthogonal to →w.

Question

1. →v=- →i+6→j
2. →w=3→i-2→j

Answer 1

In general, a projection of a vector v on the direction of a vector w is

w (w.v)/|w|²,

where (w.v) is the dot product of w and v.

If you want, let me know and I can derive it for you.

In your example,

v = -i + 6j

w = 3i - 2j

v1 = (3i - 2j) (-3 - 12)/(3² + 2²) = (3i - 2j)(-15)/13

Now for v2, first we get a vector w' perpendicular to w.

In general, if u is a vector

u = ai + bj

then

u' = -bi + aj

is a vector perpendicular to u.

In your example

w' = 2i + 3j

v2 = (2i + 3j) (-2 +18)/(2² + 3²) = (2i + 3j)(16)/13