Question

Let 4(1,-1,2), B(-2,-3,4) and C(4,2,-2) be three points in R’. Use vector method(s) to find the

(a) unit vector in the direction of 34B + AC.
(b) smaller angle between AB and AC. Correct the answer to ONE decimal place.
(c) coordinates of the point P such that it is on the line passing through 4 and B, and CP, AB are
orthogonal. (Hint: Let the coordinates of P be (x,y,z). Note that AP and AB are parallel.)
(d) equation of the plane containing 4, B and C.
(e) area of the triangle with 4, B and C as its vertices.
(f) value(s) of ¢ for which AB, AC and i+ Rj+ ck are on the same plane.
(g) orthogonal projection of i+ Rj —k onto the plane described in (d).

Answer 1

To find the unit vector in the direction of 34B + AC, first calculate the vector 34B by multiplying the scalar 34 by B and then add it to AC. Then, find the magnitude and divide each component by its magnitude to obtain the unit vector.

Step-by-step explanation:

To find the unit vector in the direction of 34B + AC, we first calculate the vector 34B by multiplying the scalar 34 by the vector B and then add it to the vector AC. The resulting vector is 34B + AC = (34)(-2,-3,4) + (4,2,-2).

Next, we find the magnitude of the vector 34B + AC by taking the square root of the sum of the squared components of the vector. Lastly, we divide each component of the vector 34B + AC by its magnitude to obtain the unit vector.