Question

Vector question, please see the attachment

Answer 1

Why are AB and MN perpendicular to each other?

Answer 2

Just reading that gave me a headache! Hopefully you will get some assistance, but it is above my pay grade.

Answer 3

can you paste the text?

Answer 4

@amistre64 lol Thank you!

Answer 5

for reference:

i = <1,0,0>
j = <0,1,0>
k = <0,0,1>

i+j+k = <1,1,1>

Answer 6

Wait :S
Let OA = i , OB = j and OC = i + j+ k (see Figure 2). Let M and N be points on the straight lines AB and
OC respectively such that AM : MB = (a :1 − a) and ON : NC = (b:1 − b) , where 0 < a <1 and 0 < b < 1 .
Suppose that MN is perpendicular to both AB and OC .

Answer 7

Okay, nothing. Thanks again

Answer 8

The second question is that why MN is the shortest distance between line OC and line AB?

Answer 9

Ok. AB = OB - OA = j - i = - i + j

MN = (a+b-1)i + (b-a)j+ b k

Dot product of AB and MN vector = 0 (if Ab is perpendicular to MN)

AB . MN = 0
(-i + j ) . ( (a+b-1)i + (b-a)j + b k)=0
= (-1)(a+b-1)( i . i) + (1)(b-a) (j.j) + 0=0

we know that i.i = j.j =1
-a-b+1+b-a=0
-2a+1=0
a=1/2

Answer 10

my proof is based on assumption that you have proved part a of the question

Answer 11

Let OA = i
OB = j
OC = i + j+ k (see Figure 2).

AM : MB = (a :1 − a)
ON : NC = (b:1 − b)




, where 0 < a <1 and 0 < b < 1 .
Suppose that MN is perpendicular to both AB and OC .