Vector Question.?

Question

Can someoneone help me to explain how to do it by given the complete solution.

1) Find a unit vector that is orthogonal to a and b where
a = 9 i - j - 2 k and b = -5 i + 2 j - 7 k .

2) Consider the line passing through the point P, whose position vector is -5 i + 10 j, and parallel to the vector -3 i - 4 j. Find a cartesian equation for the line.

3) Find the position vector of the point of intersection of lines L1, L2 with vector equations intersecting, where
L1: r = -2 i - 5 j + s (7 i + j)
L2: r = -4 i + 10 j + t (- i - 7 j)

Answer 1

1) the cross product aXb is orthogonal to both a and b

therefore the required unit vector orthogonal to both a and b would be (aXb) / |aXb|

where |aXb| refers to the magnitude.

2)If (x,y) is any point on the line then (x+5)i +(y-10)j is parallel to
-3 i - 4 j.

so that (x+5)/(-3) = (y-10)/(-4)
simplify.

3)for the first line, xi +yj = -2 i - 5 j + s (7 i + j)
so that x= -2 +7s
and y = -5+s

for the second line xi +yj = -4 i + 10 j + t (- i - 7 j)

so that x = -4 - t
and y = 10-7t

at the point of intersection, equating x and y coordinates,
-2 +7s = -4 - t

and -5+s = 10-7t

simplify and solve the two equations for s and t and substitute back into either of the two lines L1 or L2 to get the point of intersection

Answer 2

Looks as if qwert nailed it.


Doug