Question

VECTORS: HELP ME

A and B have [position vectors \(3i+2j-k\) and \(2i-j-8k\) respectively.

a. Find:
i. AB
ii. the unit vector \(u\) in the direction of BA.

b. Is \(u\) perpendicular to OA?
c. If C has osition vector \(i+j+ak\), and OC is perpendicular to \(ai-j+4k\), find \(a\).
d. If M is the midpoint of [AB], find the position vector of M.
e. Line

Answer 1

vector AB=(2i-j-8k)-(3i+2j-k)=-i-3j-7k
\[AB=\sqrt{\left( -1 \right)^2+\left( -3 \right)^2+\left( -7 \right)^2}=\sqrt{59}\]

Answer 2

i have to go now.

Answer 3

shouldn't AB be B-A?

Answer 4

i mean the one you did was the magnitude, correct?

Answer 5

vector AB=position vector of B-position vector of A

Answer 6

vector BA=i+3j+7k
unit vector u in the direction of BA\[=\frac{ i+3j+7k }{ \sqrt{1^2+3^2+7^2} }=\frac{ i+3j+7k }{ \sqrt{59} }\]

Answer 7

\[OA.AB=\left( 3i+2j-k \right).\left( -i-3j-7k \right)=3(-1)+2(-3)+(-1)(-7)\]
\[=-3-6+7=-2\pm \neq 0\]
OA is not perpendicular to AB

Answer 8

C
\[\left( i+j+ak \right).\left( ai-j+4k \right)=0\]
1(a)+1(-1)+a(4)=0
5a=1
a=1/5

Answer 9

D
P.V. of M
\[=\frac{ (3i+2j-k)+(2i-j-8k) }{ 2 }=\frac{ 5i+j-9k }{ 2 }\]

Answer 10

E.
eq. of line AB is
\[vector~ r=3i+2j-k +\lambda \left( -i-3j-7k \right)\]