Question

Let u=3ai+j-4ak and v=ai+4j+2k. a) Find the value of a if u and v are orthogonal. b) if a=2 then find a vector in the opposite direction of v and four times the magnitude of v.

Answer 1

if u and v are orthogonal
u.v=0

Answer 2

v=ai+4j+2k
if a=2
then
v=2i+4j+2k
4v=4(2i+4j+2k)
since the direction is opposite
it will -4(2i+4j+2k)

Answer 3

dear ajpricess..the answer given in book for a) a=2 or 2/5..any idea

Answer 4

for orthogonal vectors u.v=0
hence 3a^2 +4-8a=0
or 3a^2 -6a-2a+4=0
or 3a(a-2) -2(a-2)=0
or (a-2)(3a-2)=0
or a=2,2/3

Answer 5

u.v=0
So u need to find u.v.
u.v=3a*a+1*4+(-4a)*2
=3a^2+4-8a
3a^2-8a+4=0
3a^2-6a-2a+4=0
3a(a-2)-2(a-2)=0
(3a-2)(a-2)=0
3a-2=0 or a-2=0
3a=2 or a=2
a=2/3

Answer 6

unit vector in the direction opposite to v is -(2i+4j+2k)/sqrt(4+16+4) here a=2
hence a vector in the opposite direction of v and four times the magnitude of v is given by
-4(2i+4j+2k)/sqrt(24)=-8(i+2j+k) /(2*sqrt(6)) = -4(i+2j+k)/sqrt(6)

Answer 7

vat is the answer given for b @Nesh?

Answer 8

-8/sqrt24 (i+2j+k)

Answer 9

Follow @matricked's method to find b.

Answer 10

I am really sorry for messing it up. I forgot abt the idea of unit vector.

Answer 11

hi matricked and ajpricess thanks for ur help..

Answer 12

yw:)

Answer 13

welcome...