The components of vectors A and B are given as follows: Ax = 7.6 Bx = -5.1 Ay = -9.2 By = -6.8...

Question:

The components of vectors A and B are given as follows:

Ax = 7.6 Bx = -5.1

Ay = -9.2 By = -6.8

What is the magnitude of the vector difference B-A?

A) 170

B) 16

C) 13

D) 3.4

E) 3.5

Vectors in the Plane:

A vector can be identified by having a starting point and an endpoint. The subtraction of two vectors is calculated by subtracting the respective components of each vector. The magnitude is set as the length of the segment from the starting point to the endpoint. By the Pythagorean formula, the modulus (magnitude) is determined by: {eq}\left\| {\vec u} \right\| = \sqrt {{u_x}^2 + {u_y}^2} {/eq}.

Answer and Explanation:

{eq}\eqalign{ & {\text{If we have two vectors }}\,\overrightarrow A = {A_x}\hat i + {A_y}\hat j = \left\langle {{A_x},{A_y}} \right\rangle {\text{ and }}\,\overrightarrow B = {B_x}\hat i + {B_y}\hat j = \left\langle {{B_x},{B_y}} \right\rangle {\text{, }} \cr & {\text{then the vector }}\,\vec v - \vec u\,{\text{ is given by:}} \cr & \,\,\,\,\,\,\overrightarrow B - \overrightarrow A = \left\langle {{B_x} - {A_x},{B_y} - {B_x}} \right\rangle \cr & {\text{In this particular case}}{\text{, we have the vectors }}\,\overrightarrow A = \left\langle {7.6, - 9.2} \right\rangle \,{\text{ and }}\,\overrightarrow B = \left\langle { - 5.1, - 6.8} \right\rangle {\text{. }} \cr & {\text{Then:}} \cr & \,\,\,\vec u = \overrightarrow B - \overrightarrow A = \left\langle { - 5.1, - 6.8} \right\rangle - \left\langle {7.6, - 9.2} \right\rangle \cr & \,\,\,\vec u = \overrightarrow B - \overrightarrow A = \left\langle { - 5.1 - 7.6, - 6.8 + 9.2} \right\rangle \cr & \,\,\,\vec u = \overrightarrow B - \overrightarrow A = \left\langle {12.7, - 2.4} \right\rangle \cr & {\text{The magnitude of a vector }}\,\vec u = {u_x}\hat i + {u_y}\hat j = \left\langle {{u_x},{u_y}} \right\rangle {\text{ is given by: }} \cr & \,\,\,\,\,\,\,\left\| {\vec u} \right\| = \sqrt {{u_x}^2 + {u_y}^2} \cr & {\text{So:}} \cr & \,\,\,\,\,\,\,\left\| {\vec u} \right\| = \left\| {\overrightarrow B - \overrightarrow A } \right\| = \sqrt {{{\left( {12.7} \right)}^2} + {{\left( { - 2.4} \right)}^2}} \cr & {\text{Simplifying:}} \cr & \,\,\,\,\,\,\,\left\| {\overrightarrow B - \overrightarrow A } \right\| = \sqrt {167.05} \cr & \,\,\,\,\,\,\,\left\| {\overrightarrow B - \overrightarrow A } \right\| \simeq 13 \cr & {\text{Therefore}}{\text{, the magnitude of the vector difference }}\overrightarrow B - \overrightarrow A {\text{ is: }}\,\boxed{\left\| {\overrightarrow B - \overrightarrow A } \right\| \simeq 13} \cr} {/eq}


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Performing Operations on Vectors in the Plane

from Precalculus: High School

Chapter 20 / Lesson 1
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