Question

Let \(\mathbf{A}=2 \mathbf{i}+3 \mathbf{j}\) and \(\mathbf{B}=4 \mathbf{i}-4 \mathbf{j} .\) Show graphically, and find algebraically, the vectors \(-\mathbf{A}, 3 \mathbf{B}, \mathbf{A}-\mathbf{B}, \mathbf{B}+2 \mathbf{A}, \frac{1}{2}(\mathbf{A}+\mathbf{B})\).

Short answer

The vectors are: \( -\mathbf{A} = -2\mathbf{i} - 3\mathbf{j} \), \( 3\mathbf{B} = 12\mathbf{i} - 12\mathbf{j} \), \( \mathbf{A} - \mathbf{B} = -2\mathbf{i} + 7\mathbf{j} \), \( \mathbf{B} + 2\mathbf{A} = 8\mathbf{i} + 2\mathbf{j} \), \( \frac{1}{2}(\mathbf{A} + \mathbf{B}) = 3\mathbf{i} - \frac{1}{2}\mathbf{j} \).

Step-by-step solution

Understanding the Given Vectors

Given vectors are \( \mathbf{A} = 2\mathbf{i} + 3\mathbf{j} \) and \( \mathbf{B} = 4\mathbf{i} - 4\mathbf{j} \). Here, \( \mathbf{i} \) and \( \mathbf{j} \) are the unit vectors in the x and y directions, respectively.

Calculate the Vector \( -\mathbf{A} \)

To find \( -\mathbf{A} \), multiply each component of \( \mathbf{A} \) by -1. \[ -\mathbf{A} = -1 \cdot (2\mathbf{i} + 3\mathbf{j}) = -2\mathbf{i} - 3\mathbf{j} \]

Calculate the Vector \( 3\mathbf{B} \)

To find \( 3\mathbf{B} \), multiply each component of \( \mathbf{B} \) by 3. \[ 3\mathbf{B} = 3 \cdot (4\mathbf{i} - 4\mathbf{j}) = 12\mathbf{i} - 12\mathbf{j} \]

Calculate the Vector \( \mathbf{A} - \mathbf{B} \)

To find \( \mathbf{A} - \mathbf{B} \), subtract each component of \( \mathbf{B} \) from the corresponding component of \( \mathbf{A} \). \[ \mathbf{A} - \mathbf{B} = (2\mathbf{i} + 3\mathbf{j}) - (4\mathbf{i} - 4\mathbf{j}) = (2 - 4)\mathbf{i} + (3 + 4)\mathbf{j} = -2\mathbf{i} + 7\mathbf{j} \]

Calculate the Vector \( \mathbf{B} + 2\mathbf{A} \)

To find \( \mathbf{B} + 2\mathbf{A} \), first find \( 2\mathbf{A} \) and then add it to \( \mathbf{B} \).\[ 2\mathbf{A} = 2 \cdot (2\mathbf{i} + 3\mathbf{j}) = 4\mathbf{i} + 6\mathbf{j} \] Now add this to \( \mathbf{B} \): \[ \mathbf{B} + 2\mathbf{A} = (4\mathbf{i} - 4\mathbf{j}) + (4\mathbf{i} + 6\mathbf{j}) = (4 + 4)\mathbf{i} + (-4 + 6)\mathbf{j} = 8\mathbf{i} + 2\mathbf{j} \]

Calculate the Vector \( \frac{1}{2}(\mathbf{A} + \mathbf{B}) \)

To find \( \frac{1}{2}(\mathbf{A} + \mathbf{B}) \), first add \( \mathbf{A} \) and \( \mathbf{B} \), then multiply by \( \frac{1}{2} \). \[ \mathbf{A} + \mathbf{B} = (2\mathbf{i} + 3\mathbf{j}) + (4\mathbf{i} - 4\mathbf{j}) = (2 + 4)\mathbf{i} + (3 - 4)\mathbf{j} = 6\mathbf{i} - 1\mathbf{j} \]Now, multiply by \( \frac{1}{2} \):\[\frac{1}{2}(\mathbf{A} + \mathbf{B}) = \frac{1}{2}(6\mathbf{i} - 1\mathbf{j}) = 3\mathbf{i} - \frac{1}{2}\mathbf{j} \]

Key concepts

Vector Addition

Vector addition involves adding corresponding components of two vectors. For example, if we have vectors \(\mathbf{A} = 2\mathbf{i} + 3\mathbf{j}\) and \(\mathbf{B} = 4\mathbf{i} - 4\mathbf{j}\), their sum \(\mathbf{A} + \mathbf{B}\) is found by adding their i and j components separately. So, \(\mathbf{A} + \mathbf{B} = (2 + 4)\mathbf{i} + (3 - 4)\mathbf{j} = 6\mathbf{i} - 1\mathbf{j}\).
This method ensures that each directional component is addressed independently, making vector addition straightforward and systematic.

Scalar Multiplication

Scalar multiplication means multiplying a vector by a scalar (a real number). When a vector is multiplied by a scalar, like 3 multiplying \(\mathbf{B}\), each component of the vector is multiplied by the scalar. For example, given \(\mathbf{B} = 4\mathbf{i} - 4\mathbf{j}\), multiplying by 3, we get \(3\mathbf{B} = 3 \cdot (4\mathbf{i} - 4\mathbf{j}) = 12\mathbf{i} - 12\mathbf{j}\).
It's essential to apply the scalar to each component separately to maintain the vector's directional properties.

Vector Subtraction

Vector subtraction is like addition but with negatives. Subtracting vector \(\mathbf{B}\) from \(\mathbf{A}\), as in \(\mathbf{A} - \mathbf{B}\), means we subtract each component of \(\mathbf{B}\) from the corresponding component of \(\mathbf{A}\). For instance, with \(\mathbf{A} = 2\mathbf{i} + 3\mathbf{j}\) and \(\mathbf{B} = 4\mathbf{i} - 4\mathbf{j}\), the result is \(\mathbf{A} - \mathbf{B} = (2 - 4)\mathbf{i} + (3 + 4)\mathbf{j} = -2\mathbf{i} + 7\mathbf{j}\).
This operation essentially combines the steps of negating \(\mathbf{B}\)'s components and then adding the result to \(\mathbf{A}\).

Unit Vectors

Unit vectors are vectors with a magnitude of one. In two dimensions, the unit vectors are typically \(\mathbf{i}\) for the x-axis and \(\mathbf{j}\) for the y-axis. They serve as the building blocks for other vectors. Any vector can be expressed as a combination of these unit vectors. For instance, \(\mathbf{A} = 2\mathbf{i} + 3\mathbf{j}\) uses \(\mathbf{i}\) and \(\mathbf{j}\) to denote the vector's components in the x and y directions, respectively.
The importance of unit vectors lies in their standardized magnitude, facilitating the description and manipulation of vectors in various mathematical operations.