Points P and Q are located at (0, 2, 4) and (-3, 1,5). Calculate a) The position vector of P

b) The distance vector from P to Q

c) The distance between P and Q

d) A vector parallel to PQ with magnitude of 10

a) The position vector of P is .

b) The distance vector from P to Q is .

c) The distance between P and Q is .

d) A vector parallel to PQ with magnitude of 10 is .

Explanation:

a) The position vector of a point is the vector displacement from the origin to the location of the point. That is:

The position vector of P is .

b) First, we calculate the position vector of point Q:

The distance vector from P to Q is define by the following vectorial expression:

(1)

The distance vector from P to Q is .

c) There are two approaches to calculate the distance between P and Q:

First Method - Pythagorean Theorem:

Second Method - Dot Product:

(2)

The distance between P and Q is .

d) To determine a vector parallel to PQ with a given magnitude is determined by the following expression:

(3)

Where is the scale factor.

If we know that , and , then the vector is:

A vector parallel to PQ with magnitude of 10 is .