Problem 76 - Trigonometry

Question

These questions are available for instructors to help assess if you have successfully met the learning objectives for this section. If \(\mathbf{U}=3 \mathbf{i}+7 \mathbf{j}\) and \(\mathbf{V}=\mathbf{i}-4 \mathbf{j}\), find \(\mathbf{U}+2 \mathbf{V}\). a. \(\mathbf{i}+15 \mathbf{j}\) b. \(5 \mathbf{i}+3 \mathbf{j}\) c. \(10 \mathrm{i}-6 \mathrm{j}\) d. \(5 \mathbf{i}-\mathbf{j}\)

Step-by-Step Solution

Answer

The answer is d. \( 5\mathbf{i} - \mathbf{j} \).

Step 1: Understand the Vectors

We are given two vectors: \( \mathbf{U} = 3\mathbf{i} + 7\mathbf{j} \) and \( \mathbf{V} = \mathbf{i} - 4\mathbf{j} \). We need to calculate \( \mathbf{U} + 2 \mathbf{V} \).

Step 2: Multiply Vector \( \mathbf{V} \) by 2

Multiply \( \mathbf{V} = \mathbf{i} - 4\mathbf{j} \) by 2. This yields \( 2\mathbf{V} = 2(\mathbf{i} - 4\mathbf{j}) = 2\mathbf{i} - 8\mathbf{j} \).

Step 3: Add Vectors \( \mathbf{U} \) and \( 2 \mathbf{V} \)

Add \( \mathbf{U} = 3\mathbf{i} + 7\mathbf{j} \) and \( 2\mathbf{V} = 2\mathbf{i} - 8\mathbf{j} \). Calculate component-wise: \( (3\mathbf{i} + 2\mathbf{i}) + (7\mathbf{j} - 8\mathbf{j}) = 5\mathbf{i} - \mathbf{j} \).

Step 4: Choose the Correct Option

From the calculation, \( \mathbf{U} + 2\mathbf{V} = 5\mathbf{i} - \mathbf{j} \). Therefore, the answer is option d. \( 5\mathbf{i} - \mathbf{j} \).

Key Concepts

Vectors

Vectors are fundamental elements in mathematics and physics. They are quantities that have both magnitude (length) and direction, making them more versatile than scalars, which only have magnitude. Vectors are often represented in a coordinate system, typically using basis vectors like \( \mathbf{i} \) and \( \mathbf{j} \) in two-dimensions, which correspond to the unit vectors along the x-axis and y-axis respectively.
When dealing with vectors, it's important to understand that they can be added or subtracted to yield another vector. This operation is based on combining their corresponding components. For example, if you have a vector \( \mathbf{U} = 3\mathbf{i} + 7\mathbf{j} \), this means it has a component of 3 along the x-axis and 7 along the y-axis. This way of breaking vectors down helps simplify complex problems into more manageable pieces.
Vectors are not limited to two or three dimensions; they can exist in any number of dimensions, depending on the number of components they possess. Here’s what’s crucial to remember:

• Vectors have both direction and magnitude.
• They can be added by adding their respective components.
• Unit vectors like \( \mathbf{i} \) and \( \mathbf{j} \) help to break down vectors into standardized forms.

Scalar Multiplication

Scalar multiplication involves taking a vector and multiplying each of its components by a scalar (a real number). This operation changes the magnitude of the vector, but its direction remains the same, unless the scalar is negative, which reverses the direction.
Consider the vector \( \mathbf{V} = \mathbf{i} - 4\mathbf{j} \). To perform scalar multiplication by 2, each component of \( \mathbf{V} \) is multiplied by 2. This gives us \( 2\mathbf{V} = 2(\mathbf{i} - 4\mathbf{j}) = 2\mathbf{i} - 8\mathbf{j} \). Notice how the vector has the same direction as before, but each component is scaled by 2.
Scalar multiplication is a simple yet vital operation, as it allows for the adjustment of the scale of vectors, which is crucial in many applications including physics and engineering. The key points include:

• Scalar multiplication scales the vector’s magnitude.
• The direction remains unchanged unless the scalar is negative.
• This operation enables flexibility in manipulating vectors for various calculations.

Component-wise Operations

Component-wise operations are the bread and butter of vector mathematics. When adding or subtracting vectors, each component of the vectors is handled separately. This means that for two vectors \( \mathbf{U} \) and \( \mathbf{V} \), to find \( \mathbf{U} + \mathbf{V} \), you simply add the x-components together and the y-components together.
Take the vectors from the example: \( \mathbf{U} = 3\mathbf{i} + 7\mathbf{j} \) and \( 2\mathbf{V} = 2\mathbf{i} - 8\mathbf{j} \). To add them component-wise, combine the \( \mathbf{i} \) components: \( 3\mathbf{i} + 2\mathbf{i} = 5\mathbf{i} \), and the \( \mathbf{j} \) components: \( 7\mathbf{j} - 8\mathbf{j} = -\mathbf{j} \). Thus, the resulting vector is \( 5\mathbf{i} - \mathbf{j} \).
This approach of treating each component independently simplifies vector arithmetic, making it manageable and systematic. In summary:

• Each vector component is treated independently during addition or subtraction.
• Results in a new vector from the combined components.
• Facilitates clear, structured calculations, especially in multi-dimensional spaces.