a vector question

the lines l and m have vector equations
r = 2i - j + 4k + s(i + j - k) 　　　　 r = -2i + 2j + k + t(-2i + j + k)

the point P lies on l and the point Q has position vector 2i - k

(i)Given that the line PQ is perpendicular to l, find the position vector of P

(ii) Verify that Q lies on m and that PQ is perpendicular to m

Reply 1

Gaz031

rlagksquf

The point P lies on l so P has coordinates (2+p, -1+p, 4-p) for some integer p.
The direction vector of PQ is (p, -1+p, 5-p).
This direction vector is perpendicular to the direction vector of l. Hence:
(p,-1+p,5-p).(1,1,-1)=0.
Solve to find p.

(ii) Verify that Q lies on m and that PQ is perpendicular to m

Write the vector equation of m in the form r=i(...)+j(...)+k(...) then equate the coefficients of i, j and k of this with those of the position vector of Q. Using two of the equations you form solve for your parameter (t) and check this satisfies the third equation. Then Q lies on m at the value of your parameter that you've found.
For the second part, find the vector PQ and dot product it with perpendicular. Using the expression a.b=|a||b|cosθ you can then verify that they are indeed perpendicular.

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