Question

Vectors
Given OA=i+j, OB=4i-2j and OC=6i+kj, if A,B,C lie on a straight line,find the value of k.
@phi

Answer 1

can you find the direction vector form A to B?
then write down the equation of the line that contains points A and B ?

Answer 2

\[AB=3i-3j\]is it correct?

Answer 3

yes, and we could simplify the *direction* to (1, -1) (we don't care about the length)
and in this case, the arithmetic is easier.
any way, what is the equation of the line ?

Answer 4

\[AB=3(i-j)\]

Answer 5

the equation is
p0+ t* v

where p0 is a point on the line (I would use A, but B also works)
and v is the direction vector

Answer 6

\[-i-j+3(i-j)\]

Answer 7

we want the equation for any point
OA=i+j or (1,1)
vector from OA to OB is (3, -3) or 3(1, -1)
we don't care about the 3 for a *direction*, so just use (1,-1)

p= (1,1) + t (1, -1)

follow so far?

Answer 8

yes

Answer 9

t=0 gives us (1,1)
t=3 gives us (4, -2)

now we want
(1,1) + t(1, -1) = (6,k)

Answer 10

if we use the first component, we can solve for t
then use that t to find k

Answer 11

1+t=6
t=5
1-t=k
1-5=k
k=-4

Answer 12

yes,
point (5, -4) is on the line
p= (1,1) + t (1, -1)

(when t= 5)

Answer 13

**point (6,-4)

Answer 14

Thank you @phi :)