Determine whether the vectors \vec {AB} and \vec{PQ} are equivalent. A=(0,-2), \ B=(-5,-1), \...

Question:

Determine whether the vectors {eq}\vec {AB} {/eq} and {eq}\vec{PQ} {/eq} are equivalent.

{eq}A=(0,-2), \ B=(-5,-1), \ P=(3,-4), \ Q=(-2,-3) {/eq}

Vectors in 2D:

A vector is a geometrical object which has an orientation and magnitude (length). A vector can be used to describe a movement in a space from a point {eq}A {/eq} and to a point {eq}B {/eq}. When we have two points {eq}A\left( {{x_1},{y_1}} \right)\,{\text{ and }}\,B\left( {{x_2},{y_2}} \right) {/eq} the vector from A to B, {eq}\overrightarrow {AB} {/eq}, can be calculated from the components of the individual points and finding the differences in each case: {eq}\overrightarrow {AB} = \left\langle {{x_2} - {x_1},\,{y_2} - {y_1}} \right\rangle {/eq}. Vectors are considered equal if each of the components are equal.

Answer and Explanation:

{eq}\eqalign{ & {\text{We have the points }}A\left( {{x_1},{y_1}} \right) = A\left( {0, - 2} \right)\,{\text{ and }}B\left( {{x_2},{y_2}} \right) = B\left( { - 5, - 1} \right),\,{\text{ then the }} \cr & {\text{vector }}\,\overrightarrow {AB} \,{\text{ is given by:}} \cr & \,\,\,\,\overrightarrow {AB} = \left\langle {{x_2} - {x_1},\,{y_2} - {y_1}} \right\rangle \cr & {\text{Then:}} \cr & \,\,\,\,\,\overrightarrow {AB} = \left\langle { - 5 - 0,\, - 1 + 2} \right\rangle = \boxed{\left\langle { - 5,1} \right\rangle } \cr & {\text{Now}}{\text{, for the points }}P\left( {{x_1},{y_1}} \right) = P\left( {3, - 4} \right)\,{\text{ and }}Q\left( {{x_2},{y_2}} \right) = Q\left( { - 2, - 3} \right),\,{\text{ then the }} \cr & {\text{vector }}\,\overrightarrow {PQ} \,{\text{ is determined by:}} \cr & \,\,\,\,\overrightarrow {PQ} = \left\langle {{x_2} - {x_1},\,{y_2} - {y_1}} \right\rangle \cr & {\text{So:}} \cr & \,\,\,\,\,\overrightarrow {PQ} = \left\langle { - 2 - 3,\, - 3 + 4} \right\rangle = \boxed{\left\langle { - 5,1} \right\rangle } \cr & \;{\text{Therefore}}{\text{, the vectors }}\overrightarrow {AB} {\text{ and }}\overrightarrow {PQ} {\text{ are equivalent as they have the same components}}{\text{.}} \cr} {/eq}


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Vector Components: The Direction of a Vector

from UExcel Physics: Study Guide & Test Prep

Chapter 2 / Lesson 9
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