Question

The two vectors 2a + b and a -3b are perpendicular. Determine the angle between a and b if |a| = 2|b|

Answer 1

I've found that |2a+b| is 5sqrt(|b|) and that |a - 3b| = sqrt|b|, but I'm not sure where to go from there.

Answer 2

recall, if 2 vectors are perpendicular, the dot product of the two = 0

Answer 3

2a^2 - 3b^2 = 0 and

\[\sqrt{(2a)^2+b^2} = 2*\sqrt{a^2 + (-3b)^2}\]

Answer 4

so you can find a and b now, and with that , you can use the dot product or another method to find the angle between them

Answer 5

Sorry, I'm still a bit confused.

Answer 6

Why is 2a^2 - 3b^2 = 0?

Answer 7

ok, do you recall the magnitude of a vector is the square root of the sum of its components squared\[\left| R \right| =\sqrt{Rx^2 + Ry^2 + Rz^2}\]

Answer 8

and the dot product of 2 vectors is:

\[A * B = AxBx + AyBy + AzBz\]

Answer 9

Yes, of course. Oh. Ohhh of course. I'm sorry, my teacher writes all his vectors in a different form.

Answer 10

i,j,k notation?

Answer 11

no, (1, 2, 3) notation

Answer 12

oh, yeah that is a point, vectors use brackets <Ax, Ay, Az>

Answer 13

Vertically, I mean.

Answer 14

Sorry.

Answer 15

ok so lets label the 2 vectors in question Vector M and vector N

M = <2a , b>
N = <a , -3b>

Answer 16

right?

Answer 17

yup.

Answer 18

so they are perpendicular, meaning the DOT PRODUCT is zero..

\[M * N = 2a*a + (-3)b*b = 0\]

Answer 19

I get it now. Thanks so much!

Answer 20

k, you got it now?
equate the magnitudes as one is 2 times the other, then you have 2 equations with 2 unknowns a and b