Question

Consider the point P that divides the line segment joining the points A and B internally in the ratio 2:5, where A and B are given by position vectors a and b respectively.

a) Express AP in terms of a and b in its simplest form.
b) Which of the following vectors is parallel to AP ?
8a+14b

Answer 1

To express AP in terms of vectors a and b, we can use the position vector formula. AP can be represented as (2/7)(b - a). The vector 8a + 14b is parallel to AP.

Step-by-step explanation:

In order to find the expression for AP in terms of vector a and b, we need to first find the position vector of point P. The position vector of point P can be represented as r = a + t(b - a), where t represents the ratio 2:5.

So, for this case, we have r = a + (2/7)(b - a).

Therefore, AP is given by AP = r - a = (2/7)(b - a).

Now, to determine which vector is parallel to AP, we need to compare the given vector 8a + 14b with AP.

If the given vector can be written as the scalar multiple of AP, then it is parallel to AP.

In this case, 8a + 14b = 4(2a + 7b), which shows that it can indeed be expressed as the scalar multiple of AP.

Therefore, the vector 8a + 14b is parallel to AP.