/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 32 In Exercises \(21-38,\) let $$... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 32

In Exercises \(21-38,\) let $$\mathbf{u}=2 \mathbf{i}-5 \mathbf{j}, \mathbf{v}=-3 \mathbf{i}+7 \mathbf{j}, \text { and } \mathbf{w}=-\mathbf{i}-6 \mathbf{j}$$ Find each specified vector or scalar. $$3 u+4 v$$

Short Answer

Expert verified
The resultant vector, \(3\mathbf{u}+4\mathbf{v}\), is \(-6\mathbf{i} + 13\mathbf{j}\)

Step by step solution

01

Multiply Vector by Scalar

First, multiply the vectors \(\mathbf{u}\) and \(\mathbf{v}\) by the respective scalars. For \(\mathbf{u}\), multiply each component by 3, resulting in \(3\mathbf{u} = 3(2\mathbf{i}-5\mathbf{j}) = 6\mathbf{i} - 15\mathbf{j}\) Similarly for \(\mathbf{v}\), multiply each component by 4, resulting in \(4\mathbf{v} = 4(-3\mathbf{i}+7\mathbf{j}) = -12\mathbf{i} + 28\mathbf{j}\)
02

Add Vectors

Now, add these results together: \(3\mathbf{u}+4\mathbf{v} = (6\mathbf{i} - 15\mathbf{j}) + (-12\mathbf{i}+28\mathbf{j})\)
03

Simplify Expression

Combine the i and j components separately: \(3\mathbf{u}+4\mathbf{v} = (6\mathbf{i} - 12\mathbf{i}) + (-15\mathbf{j} + 28\mathbf{j}) = -6\mathbf{i} + 13\mathbf{j}\)

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Most popular questions from this chapter

Exercises \(116-118\) will help you prepare for the material covered in the next section. Use the distance formula to determine if the line segment with endpoints \((-3,-3)\) and \((0,3)\) has the same length as the line segment with endpoints \((0,0)\) and \((3,6)\)

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Use a graphing utility to graph the polar equation. $$r=2+4 \cos \theta$$

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