Let a = \langle -4,3,-4 \rangle and b = \langle 4,4,-1 \rangle. Compute: a. a+b b. a-b c. 2a d....

Question:

Let {eq}a = \langle -4,3,-4 \rangle {/eq} and {eq}b = \langle 4,4,-1 \rangle {/eq}. Compute:

a. {eq}a+b {/eq}

b. {eq}a-b {/eq}

c. {eq}2a {/eq}

d. {eq}3a+4b {/eq}

e. {eq}|a| {/eq}

Application of Vectors:

The given vectors are used for computing the given conditions. We have to compute the addition, subtraction, multiplication and the magnitude of the vectors. We have to add the vectors by using the formula, {eq}\ \displaystyle a+b = \langle a_{1}+b_{1}, \ a_{2}+b_{2}, \ a_{3}+b_{3} \rangle {/eq}.

Answer and Explanation:

We use the given vectors {eq}\displaystyle a = \langle -4,3,-4 \rangle {/eq} and {eq}\displaystyle b = \langle 4,4,-1 \rangle {/eq} to compute some given conditions.


a. Computing {eq}\displaystyle a+b {/eq}:

{eq}\begin{align*} \\ \displaystyle a+b &=\langle -4, \ 3, \ -4 \rangle + \langle 4, \ 4, \ -1 \rangle \\ \displaystyle &=\langle -4+4, \ 3+4, \ -4+(-1) \rangle \\ \displaystyle a+b &=\langle 0, \ 7, \ -5 \rangle \end{align*} {/eq}

Answer is{eq}\ \displaystyle \mathbf{\color{blue}{ a+b =\langle 0, \ 7, \ -5 \rangle }} {/eq}.


b. Computing {eq}\displaystyle a-b {/eq}:

{eq}\begin{align*} \\ \displaystyle a-b &=\langle -4, \ 3, \ -4 \rangle - \langle 4, \ 4, \ -1 \rangle \\ \displaystyle &=\langle -4-4, \ 3-4, \ -4-(-1) \rangle \\ \displaystyle a-b &=\langle -8, \ -1, \ -3 \rangle \end{align*} {/eq}

Answer is {eq}\ \displaystyle \mathbf{\color{blue}{ a-b =\langle -8, \ -1, \ -3 \rangle }} {/eq}.


c. Computing {eq}\displaystyle 2a {/eq}:

{eq}\begin{align*} \\ \displaystyle 2a &=(2) \langle -4, \ 3, \ -4 \rangle \\ \displaystyle &=\langle (2)(-4), \ (2)(3), \ (2)(-4) \rangle \\ \displaystyle 2a &=\langle -8, \ 6, \ -8 \rangle \end{align*} {/eq}

Answer is {eq}\ \displaystyle \mathbf{\color{blue}{ 2a =\langle -8, \ 6, \ -8 \rangle }} {/eq}.


d. Computing {eq}\displaystyle 3a+4b {/eq}:

{eq}\begin{align*} \\ \displaystyle 3a+4b &=(3)\langle -4, \ 3, \ -4 \rangle + (4)\langle 4, \ 4, \ -1 \rangle \\ \displaystyle &=\langle (3)(-4), \ (3)(3), \ (3)(-4) \rangle + \langle (4)(4), \ (4)(4), \ (4)(-1) \rangle \\ \displaystyle &=\langle -12, \ 9, \ -12 \rangle + \langle 16, \ 16, \ -4 \rangle \\ \displaystyle 3a+4b &=\langle -12+16, \ 9+16, \ -12+(-4) \rangle \\ \displaystyle 3a+4b &=\langle 4, \ 25, \ -16 \rangle \end{align*} {/eq}

Answer is {eq}\ \displaystyle \mathbf{\color{blue}{ 3a+4b =\langle 4, \ 25, \ -16 \rangle }} {/eq}.


e. Computing {eq}\displaystyle |a| {/eq}:

{eq}\begin{align*} \\ \displaystyle |a| &=\sqrt{(-4)^{2}+(3)^{2}+(-4)^{2}} \\ \displaystyle &=\sqrt{32+9} \\ \displaystyle |a| &=\sqrt{41} \end{align*} {/eq}

Answer is {eq}\ \displaystyle \mathbf{\color{blue}{ |a|=\sqrt{41} }} {/eq}.


Learn more about this topic:

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Vector Components: The Magnitude of a Vector

from UExcel Physics: Study Guide & Test Prep

Chapter 2 / Lesson 8
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