Problem 37 - Precalculus with Limits

Question

Vector Operations In Exercises 31-38, find (a) \(\mathbf{u}+\mathbf{v}\) . (b) \(\mathbf{u}-\mathbf{v},\) and \((\mathbf{c}) 2 \mathbf{u}-3 \mathbf{v} .\) Then sketch each resultant vector. $$\mathbf{u}=2 \mathbf{i}, \mathbf{v}=\mathbf{j}$$

Step-by-Step Solution

Answer

The results of the vector operations are: (a) \(\mathbf{u} + \mathbf{v} = 2\mathbf{i} + \mathbf{j}\) , (b) \(\mathbf{u} - \mathbf{v} = 2\mathbf{i} - \mathbf{j}\), (c) \(2\mathbf{u} - 3\mathbf{v} = 4\mathbf{i} - 3\mathbf{j}\). The sketches would show these in a Cartesian grid.

Step 1: Vector Addition

For \(\mathbf{u} + \mathbf{v}\), add the corresponding i and j components of vectors u and v. In this case, we have \(\mathbf{u} = 2\mathbf{i}, \mathbf{v} = \mathbf{j}\). Adding them together gives \(\mathbf{u} + \mathbf{v} = 2\mathbf{i} + \mathbf{j}\).

Step 2: Vector Subtraction

For \(\mathbf{u} - \mathbf{v}\), subtract the j component of v from the i component of u. In this case, that gives \(\mathbf{u} - \mathbf{v} = 2\mathbf{i} - \mathbf{j}\).

Step 3: Scalar Multiplication and Subtraction

For \(2\mathbf{u} - 3\mathbf{v}\), first multiply the vector \(\mathbf{u}\) by 2 and \(\mathbf{v}\) by 3. This gives \(2\mathbf{u} = 4\mathbf{i}\)and \(3\mathbf{v} = 3\mathbf{j}\). Then, subtract 3j from 4i, giving \(2\mathbf{u} - 3\mathbf{v} = 4\mathbf{i} - 3\mathbf{j}\)

Step 4: Sketching the Vectors

The vectors are represented on a two-dimensional plane. The i component corresponds to the x-coordinate while the j component corresponds to the y-coordinate. For \(2\mathbf{i} + \mathbf{j}\), from the origin, draw a line towards right denoting 2 units (i) and a line upwards denoting 1 unit (j). For \(2\mathbf{i} - \mathbf{j}\), from the origin, draw a line towards right denoting 2 units (i) and a line downwards denoting 1 unit. For \(4\mathbf{i} - 3\mathbf{j}\), from the origin, draw a line towards right denoting 4 units and a line downwards denoting 3 units.

Key Concepts

Vector Addition

When combining two vectors through addition, we simply align them tip-to-tail and draw a new vector from the starting point of the first vector to the end point of the second. In our example, to calculate \(\mathbf{u}+\mathbf{v}\), we add the horizontal and vertical components separately. Since \(\mathbf{u}\) has a horizontal component of 2 (represented by \(2\mathbf{i}\)) and \(\mathbf{v}\) has a vertical component of 1 (represented by \(\mathbf{j}\)), the resultant vector is a simple combination of these directions leading to \(2\mathbf{i} + \mathbf{j}\).
To visualize this, imagine moving 2 units to the right and 1 unit up from the origin on a graph. This action will lead you to the tip of the new vector formed by the addition.

Vector Subtraction

Vector subtraction, unlike addition, requires us to reverse the direction of the vector being subtracted before aligning it tip-to-tail with the other. For our exercise, subtracting \(\mathbf{v}\) from \(\mathbf{u}\) means reversing \(\mathbf{j}\) and then combining it with \(2\mathbf{i}\). The result is \(2\mathbf{i} - \mathbf{j}\), which indicates a movement 2 units to the right and 1 unit down from the origin.
Picture standing at the center of a grid and taking two steps right, followed by one step down. The spot you finish on marks the end of the vector resulting from this subtraction.

Scalar Multiplication in Vectors

To stretch or shrink a vector, we use scalar multiplication. Multiplying a vector by a number scales its length by that factor, without changing its direction. In our exercise, multiplying \(\mathbf{u}\) by 2 results in \(4\mathbf{i}\), indicating a vector twice as long in the horizontal direction. Multiplying \(\mathbf{v}\) by 3 gives \(3\mathbf{j}\), tripling its length in the vertical direction. Combining the scaled vectors through subtraction yields \(4\mathbf{i} - 3\mathbf{j}\), representing a vector that extends 4 units to the right and 3 units down from the starting point.
If you imagine stretching a rubber band outward from its midpoint, its ends move farther from the center; similarly, scaling a vector extends its reach along its original line of action.

Sketching Vectors

Drawing vectors involves placing them on a coordinate system, respecting their directional components. A vector like \(\mathbf{i}\) extends horizontally, while \(\mathbf{j}\) goes vertically. Our examples produce three distinct vectors; \(2\mathbf{i} + \mathbf{j}\) rising diagonally right and up, \(2\mathbf{i} - \mathbf{j}\) diagonally right and down, and \(4\mathbf{i} - 3\mathbf{j}\) extending significantly towards the right and down, emphasizing the effects of the scalar multiplication and subtraction operations.
Sketching these vectors on paper helps to solidify the conceptual understanding of how they interact in a plane. Starting from the origin point, each vector's path is plotted step by step to illustrate the final direction and magnitude as determined by its respective operations.