Problem 50 - Trigonometry

Question

For each pair of vectors, find \(\mathbf{U}+\mathbf{V}, \mathbf{U}-\mathbf{V}\), and \(3 \mathbf{U}+2 \mathbf{V}\). $$\mathbf{U}=5 \mathbf{i}+3 \mathbf{j}, \mathbf{V}=-3 \mathbf{i}-5 \mathbf{j}$$

Step-by-Step Solution

Answer

\(\mathbf{U}+\mathbf{V} = 2\mathbf{i} - 2\mathbf{j}\), \(\mathbf{U}-\mathbf{V} = 8\mathbf{i} + 8\mathbf{j}\), \(3\mathbf{U}+2\mathbf{V} = 9\mathbf{i} - \mathbf{j}\).

Step 1: Addition of Vectors U and V

To find \(\mathbf{U} + \mathbf{V}\), add the corresponding components of \(\mathbf{U}\) and \(\mathbf{V}\). The \(i\) component is \(5 + (-3) = 2\), and the \(j\) component is \(3 + (-5) = -2\). So, \(\mathbf{U} + \mathbf{V} = 2\mathbf{i} - 2\mathbf{j}\).

Step 2: Subtraction of Vectors U and V

To find \(\mathbf{U} - \mathbf{V}\), subtract the components of \(\mathbf{V}\) from \(\mathbf{U}\). The \(i\) component is \(5 - (-3) = 5 + 3 = 8\), and the \(j\) component is \(3 - (-5) = 3 + 5 = 8\). So, \(\mathbf{U} - \mathbf{V} = 8\mathbf{i} + 8\mathbf{j}\).

Step 3: Scalar Multiplication and Addition

To find \(3\mathbf{U} + 2\mathbf{V}\), first calculate \(3\mathbf{U}\) and \(2\mathbf{V}\). For \(3\mathbf{U}\), multiply each component of \(\mathbf{U}\) by 3: \(3(5\mathbf{i} + 3\mathbf{j}) = 15\mathbf{i} + 9\mathbf{j}\). For \(2\mathbf{V}\), multiply each component of \(\mathbf{V}\) by 2: \(2(-3\mathbf{i} - 5\mathbf{j}) = -6\mathbf{i} - 10\mathbf{j}\). Now add these results: \((15\mathbf{i} + 9\mathbf{j}) + (-6\mathbf{i} - 10\mathbf{j}) = (15 - 6)\mathbf{i} + (9 - 10)\mathbf{j} = 9\mathbf{i} - \mathbf{j}\).

Key Concepts

Vector Subtraction

Vector subtraction is a way to find the difference between two vectors. Imagine you're trying to figure out how much one force can counteract another. This is what vector subtraction helps with. In simple terms, to subtract one vector from another, you subtract their corresponding components.
To break it down:

• Take the first vector, which we'll call \( \mathbf{U} = 5 \mathbf{i} + 3 \mathbf{j} \).
• The second vector is \( \mathbf{V} = -3 \mathbf{i} - 5 \mathbf{j} \).

Subtraction means we take each component from \( \mathbf{V} \) and subtract it from the respective component in \( \mathbf{U} \).

• The \( i \)-component: \( 5 - (-3) = 5 + 3 = 8 \).
• The \( j \)-component: \( 3 - (-5) = 3 + 5 = 8 \).

This calculation shows that \( \mathbf{U} - \mathbf{V} = 8\mathbf{i} + 8 \mathbf{j} \). By understanding this process, you can visualize how forces or displacements act against each other. Simple, right? Just take each component and perform straightforward subtraction.

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying a vector by a real number, known as a scalar. This operation scales or "stretches" the vector without changing its direction unless the scalar is negative, in which case it also reverses the direction.
Here's how it works:

• Given a vector \( \mathbf{U} = 5 \mathbf{i} + 3 \mathbf{j} \) and a scalar, say 3, scalar multiplication looks like this: \( 3 \times \mathbf{U} = 3(5 \mathbf{i} + 3 \mathbf{j}) = 15 \mathbf{i} + 9 \mathbf{j} \).
• For another vector \( \mathbf{V} = -3 \mathbf{i} - 5 \mathbf{j} \) and a scalar 2, the operation is: \( 2 \times \mathbf{V} = 2(-3 \mathbf{i} - 5 \mathbf{j}) = -6 \mathbf{i} - 10 \mathbf{j} \).

The result of scalar multiplication is a vector that maintains the original vector's direction but has a magnitude altered by the scalar factor. Don't forget: if the scalar is positive, it extends the vector along the same direction. If negative, it flips the vector's direction.

Components of a Vector

Understanding the components of a vector is key in vector mathematics. Vectors are often described by their components along the coordinate axes, typically the \( i \)-component (horizontal) and the \( j \)-component (vertical), in a 2D space.
Each vector is imagined as having two parts:

• The \( i \)-component, which moves horizontally, is like drawing a straight line left or right on a graph. For example, in vector \( \mathbf{U} = 5 \mathbf{i} + 3 \mathbf{j} \), 5 is the horizontal component.
• The \( j \)-component, which moves vertically, is like drawing a line up or down. Here, 3 is the vertical component.

When you combine these two components, you can precisely describe the vector's overall movement. This makes it very easy to perform operations like addition and subtraction on vectors since you handle each component separately. Just imagine each piece as a distinct axis movement, coordinate by coordinate. This decomposition is crucial for understanding how vectors work in physics and engineering.