Question

I'm having difficulty with orthogonal vectors, can anyone help me out?
Find "a" so that the vectors v = i + aj and w = -4i - 2j are orthogonal.

Answer 1

define orthogonal ....

Answer 2

in slopes, we have orthogonal slopes (perpendicular) if one is the negative reciprocal of the other

-4, -2 would be ortho to 2, -4

but 2,-4 isnt 1,a is it .. it is if we scale it :) divide by 2

1,-2 = 1,a

Answer 3

otherwise dot and solve for a ...

Answer 4

the orthogonal should be perpendicular.

Answer 5

right

and perpendicular slopes have a product of -1

a consequence to this is: if (a,b) and (x,y) are perpendular

\[\frac ba \frac yx=-1\]
\[\frac {by}{ax} =-1\]
\[by =-ax\]
\[by+ax=0\]

Answer 6

the last line is just the dotproduct of the 2 vectors

Answer 7

a, b
. x, y
-------
ax+by

Answer 8

so in this case it would be: \[\frac{ a}{ 1}\frac{ 2 }{ 4} = -1\]

Answer 9

4+2a = -1?

Answer 10

v = i + aj and w = -4i - 2j

a/1 and 2/4

a/2 = -1

Answer 11

2a = -4

2a + 4 = 0

Answer 12

\[\frac{ a }{ 2}=-1\]
a = -2?

Answer 13

yep

Answer 14

the proper vector approach is to set the dotproduct to 0

1, a
-4,-2
-----
-4-2a = 0

2a = -4

a = -4/2

Answer 15

And that is it. wow, sometimes I way overthink this and make it too complicated.

Answer 16

Thank you for your help! My finals week will be a little lighter thanks to you!

Answer 17

good luck :)