Find the vector PM, given that the position vectors of P and Q are p and q respectively, and M is the…

Question

Find the vector PM, given that the position vectors of P and Q are p and q respectively, and M is the midpoint of PQ.

ScanSolve AI Tutor · Accepted Answer

Here is the solution to part 7 (b) (i):

7. (b) (i) Find $\vec{PM}$

Identify the given position vectors.
The position vector of P from O is $\vec{OP} = \mathbf{p}$ .
The position vector of Q from O is $\vec{OQ} = \mathbf{q}$ .
Find the vector $\vec{PQ}$ .
The vector from P to Q can be expressed as the difference between the position vector of Q and the position vector of P:
$\vec{PQ} = \vec{OQ} - \vec{OP}$ $\vec{PQ} = \mathbf{q} - \mathbf{p}$
Use the information that M is the midpoint of PQ.
If M is the midpoint of PQ, then the vector $\vec{PM}$ is half of the vector $\vec{PQ}$ .
$\vec{PM} = \frac{1}{2} \vec{PQ}$
Substitute the expression for $\vec{PQ}$ into the equation for $\vec{PM}$ .
$\vec{PM} = \frac{1}{2} (\mathbf{q} - \mathbf{p})$
The vector $\vec{PM}$ is $\boxed{\frac{1}{2}(\mathbf{q} - \mathbf{p})}$ .

Find the vector PM, given that the position vectors of P and Q are p and q respectively, and M is the midpoint of PQ.