Problem 36 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The resultant vectors are (a) \(\mathbf{u} + \mathbf{v} = -2\mathbf{i} + 4\mathbf{j}\), (b) \(\mathbf{u} - \mathbf{v} = -2\mathbf{i} - 2\mathbf{j}\), (c) $2\mathbf{u} - 3\mathbf{v} = -4\mathbf{i} - 7\mathbf{j}$. These are represented on the 2D plane starting from origin.

Step 1: Calculate the Vector Addition

First, find the sum of the vectors \(\mathbf{u}\) and \(\mathbf{v}\). The sum of two vectors can be determined by adding their corresponding components. \(\mathbf{u} + \mathbf{v}\) = \((-2\mathbf{i} + \mathbf{j}) + 3\mathbf{j}\) = \(-2\mathbf{i} + 4\mathbf{j}\)

Step 2: Calculate the Vector Subtraction

Next, find the difference of the vectors \(\mathbf{u}\) and \(\mathbf{v}\). This operation can be done by subtracting the corresponding components of \(\mathbf{v}\) from \(\mathbf{u}\). \(\mathbf{u} - \mathbf{v}\) = \((-2\mathbf{i} + \mathbf{j}) - 3\mathbf{j}\) = \(-2\mathbf{i} - 2\mathbf{j}\)

Step 3: Calculate Scalar Multiplication and Vector Subtraction

We will calculate 2\(\mathbf{u}\) - 3\(\mathbf{v}\). Multiply each component of \(\mathbf{u}\) by 2 and of \(\mathbf{v}\) by 3. Then subtract the product of 3 and \(\mathbf{v}\) from the product of 2 and \(\mathbf{u}\). So, \(2\mathbf{u} - 3\mathbf{v}\) = \(2(-2\mathbf{i} + \mathbf{j}) - 3 (3\mathbf{j})\) = \(-4\mathbf{i} + 2\mathbf{j} - 9\mathbf{j}\) = \(-4\mathbf{i} - 7\mathbf{j}\)

Step 4: Vector Representation

Draw these vectors on the coordinate plane. Remember to start drawing from the origin (0,0). The vector \(\mathbf{i}\) shows movement along x-axis and \(\mathbf{j}\) along y-axis. Negative sign shows movement in opposite direction.

Key Concepts

Vector Addition

Vector addition is an essential concept in mathematics, where two vectors are combined to produce a third vector that represents the combined effect of the two. It's much like combining forces: if one force pushes a cart north and another pushes it east, the cart moves northeast. In more technical terms,

• Add the corresponding components of the vectors together.
• For example, if you have vectors \(-2\mathbf{i} + \mathbf{j}\) and \(+3\mathbf{j}\), their sum is obtained by adding their individual components.
• The result will be \(-2\mathbf{i} + 4\mathbf{j}\).

This is because although \(-2\mathbf{i}\) has no component in \(+3\mathbf{j}\), it's summed directly to \(+3\mathbf{j}\), creating the combined effect along the y-direction and retaining movement in the x-direction.

Vector Subtraction

Vector subtraction is like vector addition, but instead of adding, we subtract components of one vector from the other. This essentially finds the difference between two vectors:

• Subtract the corresponding components of one vector from the other.
• Consider vectors \(-2\mathbf{i} + \mathbf{j}\) and \(+3\mathbf{j}\).
• Here, \(-2\mathbf{i}\) remains the same because it doesn't interact with the j-components, while the j-components subtract to \(-2\mathbf{j}\).
• This gives a resultant vector of \(-2\mathbf{i} - 2\mathbf{j}\).

This subtraction gives a vector that indicates the difference in direction and magnitude between the initial vectors.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a real number), which alters the magnitude of the vector without changing its direction unless the scalar is negative. If the scalar is positive, the vector stretches; if negative, it reverses direction:

• Apply the scalar to each component of the vector independently.
• For instance, multiplying \-2\mathbf{i} + \mathbf{j}\ by 2 gives \-4\mathbf{i} + 2\mathbf{j}\.
• If you've another vector \(+3\mathbf{j}\) and multiply by 3, you get \(+9\mathbf{j}\).

To calculate \(2u - 3v\), perform these multiplications and subtract, resulting in \(-4\mathbf{i} - 7\mathbf{j}\). This showcases how scalars change vector magnitudes and affect operations like addition or subtraction.

Coordinate System

The coordinate system provides a framework for visualizing vectors. Typically, the Cartesian coordinate system is used, which consists of two axes: x-axis (horizontal) and y-axis (vertical). Vectors are represented in this system using unit vectors:

• The vector \(-2\mathbf{i}\) indicates movement along the negative x-axis.
• Likewise, \(+4\mathbf{j}\) would show movement along the y-axis.
• It helps to start drawing vectors from the origin (0,0), and extend components along the specified axes.
• Remember, signs inform direction: positive moves right/up, negative moves left/down.

This system simplifies the visualization of vector operations and helps in understanding the impact of vector math like addition and subtraction on direction and magnitude.