If P = (-1,1) and Q=(3,2), find the components of \vec {PQ}

Question:

If {eq}P = (-1,1) {/eq} and {eq}Q=(3,2) {/eq}, find the components of {eq}\vec {PQ} {/eq}

Vectors in 2D:

A vector is a geometrical shape that is similar to a line but has an orientation and is characterized by a starting point and an endpoint, that is, a point and a {eq}P\left( {{x_1},{y_1}} \right)\,{\text{ and }}\,Q\left( {{x_2},{y_2}} \right) {/eq}. When we have two points {eq}P\left( {{x_1},{y_1}} \right)\,{\text{ and }}\,Q\left( {{x_2},{y_2}} \right) {/eq} we can establish a vector in which the components are the differences in the coordinates of the two points: {eq}\vec v = \overrightarrow {PQ} = \left\langle {{x_2} - {x_1},\,{y_2} - {y_1}} \right\rangle = \left( {{x_2} - {x_1}} \right)\hat i + \left( {{y_2} - {y_1}} \right)\hat j {/eq}.

Answer and Explanation:

{eq}\eqalign{ & {\text{We have the points }}P\left( {{x_1},{y_1}} \right) = P\left( { - 1,1} \right)\,{\text{ and }}Q\left( {{x_2},{y_2}} \right) = Q\left( {3,2} \right),\,{\text{ then the }} \cr & {\text{vector }}\,\overrightarrow {PQ} \,{\text{ is given by:}} \cr & \,\,\,\,\overrightarrow {PQ} = \left\langle {{x_2} - {x_1},\,{y_2} - {y_1}} \right\rangle \cr & {\text{Thus}}{\text{, in this particular case we have:}} \cr & \,\,\,\,\,\overrightarrow {PQ} = \left\langle {3 - \left( { - 1} \right),\,2 - 1} \right\rangle \cr & {\text{Simplifying:}} \cr & \,\,\,\,\,\overrightarrow {PQ} = \left\langle {3 + 1,\,2 - 1} \right\rangle = \left\langle {4,1} \right\rangle \cr & \;{\text{Since}}{\text{, the standard equivalent representation of the vector }}\,\overrightarrow {PQ} \,{\text{ is:}} \cr & \,\,\,\,\,\vec v = \overrightarrow {PQ} = 4\hat i + \hat j \cr & {\text{Therefore}}{\text{, the components of }}\vec v = \overrightarrow {PQ} {\text{ are: }}\boxed{{v_x} = 4,\,\,{v_y} = 1} \cr} {/eq}


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