# Write down vector equations of the lines AD and BC and find the position vector of the point of intersection.Points A,B,C,D in a plane have position vectors a = 6i + 8j, b = 3/2a, c = 6i + 3j, d = 5/3c respectively. Write down vector equations of the lines AD and BC and find the position vector of the point of intersection.   I found AD to be (6i + 8j) + t(4i - 3j) which is right,   But I cant find BC so any help would be much appreciated. I used the formula r = c + t(d-c) to find AD. The answer for BC is (9i + 12j) +s(i + 3j).   Have also found the point of intersection which is right and is 22/3i + 7j   Thankyou so much The position vectors of points A,B,C,D are,

`hata = 6i+8j`

`hatb = 3/2a` ----> `hatb = 3/2(6i+8j) = 9i+12j`

`hatc = 6i+3j`

`hatd = 5/3c`   -----> `hatd = 5/3(6i+3j) = 10i+5j`

`r = a+t(d-a)`

`r = (6i+8j)+t((10i+5j)-(6i+8j))`

`r = (6i+8j)+t(4i-3j)`

Simplifyng,

`r = (6+4t)i+(8-3t)j`

Line BC is given by,

`p = b+s(c-b)`

`p = (9i+12j)+s((6i+3j)-(9i+12j))`

`p = (9i+12j)+s(-3i-9j)`

`P = (9i+12j)-s(3i+9j)`

Simplifying,

`p = (9-3s)i+(12-9s)j`

Now if these two lines, intersect each other,

`(6+4t)i+(8-3t)j = (9-3s)i+(12-9s)j`

The two i, j components on either side must be equal to each other separately.

Therefore,

`6+4t = 9-3s`    and   `8-3t = 12-9s`

This gives two equations to find t and s.

`4t+3s = 3`    and    `3t-9s = -4`

Solving these two equations you would get,

`t = 1/3`       and   `s = 5/9`

so the point of interesection can be found by substituting t value in AD line,

If the point of intersection is E,

`hate = (6+4(1/3))i+(8-3(1/3))j`

`hate = 22/3i+7j`

The point of intesection of AD and BC  is (22/3)i+7j.