(3) Given that P- = (2/3) and Q- = - (1/4), find the coordinates of P

Question

(3) Given that P-> = (2/3) and Q-> = - (1/4), find the coordinates of P

ScanSolve AI Tutor · Accepted Answer

let's explain question (3) again.

3) Given $\vec{PQ} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $\vec{OQ} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}$ . Find the coordinates of P.

Understand the relationship between position vectors and displacement vectors.
A displacement vector from point P to point Q, denoted as $\vec{PQ}$ , can be found by subtracting the position vector of the starting point P ( $\vec{OP}$ ) from the position vector of the ending point Q ( $\vec{OQ}$ ).
So, the fundamental relationship is:
$\vec{PQ} = \vec{OQ} - \vec{OP}$
Our goal is to find the coordinates of P, which are represented by the position vector $\vec{OP}$ . We need to rearrange the equation to solve for $\vec{OP}$ .
Add $\vec{OP}$ to both sides and subtract $\vec{PQ}$ from both sides:
$\vec{OP} = \vec{OQ} - \vec{PQ}$
Substitute the given vectors into the rearranged equation.
We are given $\vec{OQ} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}$ and $\vec{PQ} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ .
$\vec{OP} = \begin{pmatrix} -1 \\ 4 \end{pmatrix} - \begin{pmatrix} 2 \\ 3 \end{pmatrix}$
Perform the vector subtraction.
To subtract vectors, you subtract their corresponding components (x-component from x-component, y-component from y-component).
$\vec{OP} = \begin{pmatrix} -1 - 2 \\ 4 - 3 \end{pmatrix}$
Calculate the resulting components.
$\vec{OP} = \begin{pmatrix} -3 \\ 1 \end{pmatrix}$
State the coordinates of P.
The position vector $\vec{OP} = \begin{pmatrix} -3 \\ 1 \end{pmatrix}$ means that point P has coordinates $(-3, 1)$ .

(3) Given that P-> = (2/3) and Q-> = - (1/4), find the coordinates of P