Problem 45 - Trigonometry

Question

For each pair of vectors, find \(\mathbf{U}+\mathbf{V}, \mathbf{U}-\mathbf{V}\), and \(3 \mathbf{U}+2 \mathbf{V}\). $$\mathbf{U}=-\mathbf{i}+\mathbf{j}, \mathbf{V}=\mathbf{i}+\mathbf{j}$$

Step-by-Step Solution

Answer

\( \mathbf{U} + \mathbf{V} = 2\mathbf{j} \), \( \mathbf{U} - \mathbf{V} = -2\mathbf{i} \), \( 3\mathbf{U} + 2\mathbf{V} = -\mathbf{i} + 5\mathbf{j} \)

Step 1: Understanding Vectors

The given vectors are \( \mathbf{U} = -\mathbf{i} + \mathbf{j} \) and \( \mathbf{V} = \mathbf{i} + \mathbf{j} \). These vectors are expressed in terms of unit vectors \( \mathbf{i} \) and \( \mathbf{j} \), which represent the x and y directions, respectively.

Step 2: Calculating \(\mathbf{U} + \mathbf{V}\)

To add the vectors \(\mathbf{U} = -\mathbf{i} + \mathbf{j}\) and \(\mathbf{V} = \mathbf{i} + \mathbf{j}\), sum their respective components: \((-\mathbf{i} + \mathbf{i}) + (\mathbf{j} + \mathbf{j}) = 0\mathbf{i} + 2\mathbf{j}\). Thus, \(\mathbf{U} + \mathbf{V} = 0\mathbf{i} + 2\mathbf{j}\).

Step 3: Calculating \(\mathbf{U} - \mathbf{V}\)

To subtract vector \(\mathbf{V}\) from \(\mathbf{U}\), subtract their corresponding components: \((-\mathbf{i} - \mathbf{i}) + (\mathbf{j} - \mathbf{j}) = -2\mathbf{i} + 0\mathbf{j}\). Therefore, \(\mathbf{U} - \mathbf{V} = -2\mathbf{i} + 0\mathbf{j}\).

Step 4: Calculating \(3\mathbf{U} + 2\mathbf{V}\)

First, find the scalar multiplication for each vector: \(3\mathbf{U} = 3(-\mathbf{i} + \mathbf{j}) = -3\mathbf{i} + 3\mathbf{j}\) and \(2\mathbf{V} = 2(\mathbf{i} + \mathbf{j}) = 2\mathbf{i} + 2\mathbf{j}\). Then, add these results: \((-3\mathbf{i} + 3\mathbf{j}) + (2\mathbf{i} + 2\mathbf{j}) = (-3\mathbf{i} + 2\mathbf{i}) + (3\mathbf{j} + 2\mathbf{j}) = -\mathbf{i} + 5\mathbf{j}\). Hence, \(3\mathbf{U} + 2\mathbf{V} = -\mathbf{i} + 5\mathbf{j}\).

Key Concepts

Vector Addition

Vector addition is a fundamental operation that lets you combine two vectors to get another vector. Visualize each vector as an arrow pointing in a specific direction with a certain length. Adding vectors means arranging these arrows tip-to-tail and drawing a new vector from the start of the first vector to the end of the second.
This graphical method makes it easier to understand how vectors combine. However, mathematically, vector addition is performed by adding the corresponding components of the vectors.

• For vectors \( \mathbf{U} = -\mathbf{i} + \mathbf{j} \) and \( \mathbf{V} = \mathbf{i} + \mathbf{j} \), the x-components are \(-\mathbf{i}\) and \(\mathbf{i}\), while the y-components are both \(\mathbf{j}\).
• Adding these components: \( (-\mathbf{i} + \mathbf{i}) + (\mathbf{j} + \mathbf{j}) = 0\mathbf{i} + 2\mathbf{j} \).

Thus, the sum \( \mathbf{U} + \mathbf{V} \) results in a vector pointing straight up along the y-axis with a length equal to twice that of a single unit vector \( \mathbf{j} \). This makes it clear why the resultant vector \( 0\mathbf{i} + 2\mathbf{j} \) has no x-component.

Vector Subtraction

Subtraction of vectors is similar to addition, but with a small twist. Subtracting vectors involves reversing the direction of the vector you are subtracting, and then adding it to the other vector.
This approach essentially converts vector subtraction into vector addition with an opposite-direction vector.

• For our given vectors \( \mathbf{U} = -\mathbf{i} + \mathbf{j} \) and \( \mathbf{V} = \mathbf{i} + \mathbf{j} \), we subtract them by reversing \( \mathbf{V} \).
• The x-components become \( -\mathbf{i} - \mathbf{i} \), and the y-components become \( \mathbf{j} - \mathbf{j} \).
• This gives \( -2\mathbf{i} + 0\mathbf{j} \), which indicates a vector pointing directly left along the x-axis.

Hence, \( \mathbf{U} - \mathbf{V} = -2\mathbf{i} + 0\mathbf{j} \) captures a clear directional shift purely on the x-axis with no y component.

Scalar Multiplication

Scalar multiplication refers to scaling a vector by a real number called a "scalar". This process changes the magnitude of the vector but not its direction unless the scalar is negative.This multiplication affects each component of the vector by the scalar amount.
For instance, given vectors \( \mathbf{U} = -\mathbf{i} + \mathbf{j} \) and \( \mathbf{V} = \mathbf{i} + \mathbf{j} \):

• Multiplying \( \mathbf{U} \) by 3 results in \( 3\mathbf{U} = 3(-\mathbf{i} + \mathbf{j}) = -3\mathbf{i} + 3\mathbf{j} \).
• Similarly, \( 2\mathbf{V} = 2(\mathbf{i} + \mathbf{j}) = 2\mathbf{i} + 2\mathbf{j} \).

Once scaled, you can combine these results as follows:

• Add \( -3\mathbf{i} + 3\mathbf{j} \) and \( 2\mathbf{i} + 2\mathbf{j} \) by summing their respective components to get \( (-3\mathbf{i} + 2\mathbf{i}) + (3\mathbf{j} + 2\mathbf{j}) = -\mathbf{i} + 5\mathbf{j} \).

This shows how scalar multiplication can stretch or shrink a vector while the combination of scaled vectors provides a new direction and magnitude, illustrated by \( 3\mathbf{U} + 2\mathbf{V} = -\mathbf{i} + 5\mathbf{j} \).