Find a vector u that is parallel to the yzplane and perpendi

Find a vector u that is parallel to the yz-plane and perpendicular to the vector v = (3, 3, 5). Find unit vectors that satisfy the given conditions: The unit vector in the same direction as (1, -5) is The unit vector oppositely directed to -2i + 3j - 3k is The unit vector that has the same direction as the vector from the point A = (-5, 5) to the point B = (-4, 5).

Solution

Vectors parallel to the yz-plane are those with zero component in the i direction, and so have the form

u = (0, , ). where and are arbitrary real numbers.

We seek a vector which is also perpendicular to v = (3, 3, 5),

so we must have u · v = 0, which means that we will need

(0, , ).(3,3,5) = 0.

The latter requirement is equivalent to +3 + 5 = 0,

3 + 5 = 0,

3 = -5

= -5/3

= -3/5

and this gives =(-3/5) . We can therefore take u = (0, , -3/5) , where is any real number.