Problem 32 - Algebra and Trigonometry

Question

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. $$3 u+4 v$$

Step-by-Step Solution

Answer

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

Step 1: 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 \(4\mathbf{v} = 4(-3\mathbf{i}+7\mathbf{j}) = -12\mathbf{i} + 28\mathbf{j}\)

Step 2: Add Vectors

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

Step 3: Simplify Expression

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