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Chapter 12: Problem 33

Let \(\mathbf{u}=\langle 4,-2\rangle, \mathbf{v}=\langle-4,6\rangle,\) and \(\mathbf{w}=\langle 0,8\rangle .\) Express the following vectors in the form \(\langle a, b\rangle\) $$8 w+v-6 u$$

Short Answer

Expert verified
Question: Determine the vector resulting from the operation \(8\mathbf{w}+\mathbf{v}-6\mathbf{u}\) given the following vectors: \(\mathbf{u}=\langle 4,-2\rangle\), \(\mathbf{v}=\langle -4, 6\rangle\), and \(\mathbf{w}=\langle 0, 8\rangle\). Answer: The resulting vector is \(\langle -28, 82 \rangle\).

Step by step solution

01

Multiply vectors by scalars

We first need to multiply each vector by their respective scalar: $$6\mathbf{u} = 6\langle 4,-2\rangle = \langle 6(4), 6(-2)\rangle = \langle 24, -12\rangle$$ $$\mathbf{v} = \langle -4, 6\rangle$$ $$8\mathbf{w} = 8\langle 0, 8\rangle = \langle 8(0), 8(8)\rangle = \langle 0, 64\rangle$$
02

Add and subtract the resulting vectors

Now, we need to perform the given operation on these resulting vectors: \(8\mathbf{w}+\mathbf{v}-6\mathbf{u}\). Let's do this step by step: $$\langle 0, 64 \rangle + \langle -4, 6 \rangle - \langle 24, -12 \rangle = \langle 0-4, 64+6 \rangle + \langle -24, 12 \rangle = \langle -4, 70 \rangle + \langle -24, 12 \rangle$$ Now, we can add the two vectors: $$\langle -4, 70 \rangle + \langle -24, 12 \rangle = \langle -4-24, 70+12 \rangle = \langle -28, 82 \rangle$$
03

Write the final vector in the form \(\langle a, b \rangle\)

The final vector resulting from the operation \(8\mathbf{w}+\mathbf{v}-6\mathbf{u}\) is: $$\langle -28, 82 \rangle$$

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