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Chapter 12: Problem 30

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 \(\langle a, b\rangle\) $$2 \mathbf{u}+3 \mathbf{v}$$

Short Answer

Expert verified
Question: Express the vector \(2 \mathbf{u}+3 \mathbf{v}\) in the form of \(\langle a, b \rangle\), where \(\mathbf{u}=\langle 4, -2 \rangle\) and \(\mathbf{v}=\langle -4, 6 \rangle\). Answer: The vector \(2 \mathbf{u} + 3 \mathbf{v}\) in the form of \(\langle a, b \rangle\) is \(\langle -4, 14 \rangle\).

Step by step solution

01

Calculate the scalar multiple of each vector

Multiply each vector by their respective scalar values: 2 \(\mathbf{u} = 2\langle 4, -2 \rangle = \langle 8, -4 \rangle\) 3 \(\mathbf{v} = 3\langle -4, 6 \rangle = \langle -12, 18 \rangle\)
02

Add the resulting vectors together

Add the two resulting vectors from step 1: \(\langle 8, -4 \rangle + \langle -12, 18 \rangle = \langle 8 - 12, -4 + 18 \rangle\)
03

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

Perform the calculations inside the brackets: \(\langle -4, 14 \rangle\) The vector \(2 \mathbf{u} + 3 \mathbf{v}\) in the form of \(\langle a, b \rangle\) is \(\boxed{\langle -4, 14 \rangle}\).

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