Problem 48 - Trigonometry

Question

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

Step-by-Step Solution

Answer

\(\mathbf{U} + \mathbf{V} = -3\mathbf{i} + 5\mathbf{j}\), \(\mathbf{U} - \mathbf{V} = -3\mathbf{i} - 5\mathbf{j}\), \(3\mathbf{U} + 2\mathbf{V} = -9\mathbf{i} + 10\mathbf{j}\).

Step 1: Understand the Vectors

The vector \(\mathbf{U}\) is given as \(-3 \mathbf{i}\) which means it points in the negative direction of the x-axis. The vector \(\mathbf{V}\) is \(5 \mathbf{j}\), indicating it points in the positive direction of the y-axis.

Step 2: Add the Vectors \(\mathbf{U} + \mathbf{V}\)

To find \(\mathbf{U} + \mathbf{V}\), simply add the corresponding components. Since \(\mathbf{U} = -3\mathbf{i} + 0\mathbf{j}\) and \(\mathbf{V} = 0\mathbf{i} + 5\mathbf{j}\), we have: \(-3 + 0\) for the \(\mathbf{i}\) components and \(0 + 5\) for the \(\mathbf{j}\) components. This results in \(-3\mathbf{i} + 5\mathbf{j}\).

Step 3: Subtract the Vectors \(\mathbf{U} - \mathbf{V}\)

Subtract the vector \(\mathbf{V}\) from \(\mathbf{U}\) by subtracting the corresponding components. Therefore, \(\mathbf{U} - \mathbf{V} = (-3 - 0)\mathbf{i} + (0 - 5)\mathbf{j} = -3\mathbf{i} - 5\mathbf{j}\).

Step 4: Find the Scalar Multiplication \(3\mathbf{U} + 2\mathbf{V}\)

First, multiply each vector by its respective scalar. This gives: \(3\mathbf{U} = 3(-3\mathbf{i}) = -9\mathbf{i}\) and \(2\mathbf{V} = 2(5\mathbf{j}) = 10\mathbf{j}\). Then add these results: \(-9\mathbf{i} + 0\mathbf{j} + 0\mathbf{i} + 10\mathbf{j} = -9\mathbf{i} + 10\mathbf{j}\).

Key Concepts

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a constant real number). This process changes the magnitude of the vector but not its direction, unless the scalar is negative, which reverses the direction. For a vector \( \mathbf{A} = a\mathbf{i} + b\mathbf{j} \), multiplying by a scalar \( c \) is done by:

• Multiplying each component by \( c \)

Thus, \( c\mathbf{A} = ca\mathbf{i} + cb\mathbf{j} \). Let’s take an example with our vector \( \mathbf{U} = -3\mathbf{i} \): a scalar multiplication by \( 3 \) yields \( 3 \mathbf{U} = 3(-3\mathbf{i}) = -9\mathbf{i} \). Similarly, for \( \mathbf{V} = 5\mathbf{j} \), multiplying by \( 2 \) gives \( 2\mathbf{V} = 2(5\mathbf{j}) = 10\mathbf{j} \). These newly-scaled vectors can then be used in further vector operations, like addition.

Vector Subtraction

Vector subtraction involves reversing the direction of the vector being subtracted and then adding it to the initial vector. Essentially, if you have vectors \( \mathbf{U} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{V} = c\mathbf{i} + d\mathbf{j} \), then the operation \( \mathbf{U} - \mathbf{V} \) can be thought of as \( \mathbf{U} + (-\mathbf{V}) \), where \( -\mathbf{V} = -c\mathbf{i} - d\mathbf{j} \).

• This means subtracting each corresponding component of \( \mathbf{V} \) from \( \mathbf{U} \):

\( \mathbf{U} - \mathbf{V} = (a-c)\mathbf{i} + (b-d)\mathbf{j} \). In our specific example, since \( \mathbf{U} = -3\mathbf{i} \) and \( \mathbf{V} = 5\mathbf{j} \), we do \( \mathbf{U} - \mathbf{V} = (-3 - 0)\mathbf{i} + (0 - 5)\mathbf{j} = -3\mathbf{i} - 5\mathbf{j} \). Notice how subtraction effectively adds the "opposite" of vector \( \mathbf{V} \).

Unit Vectors

Unit vectors are vectors with a magnitude of exactly 1 unit. They are particularly useful in defining directions in Cartesian coordinates. Commonly, unit vectors are represented as \( \mathbf{i} \) and \( \mathbf{j} \), which point in the direction of the x-axis and y-axis, respectively.

• These vectors form the basis for representing other vectors in the 2D plane:
• Any vector \( \mathbf{A} = a\mathbf{i} + b\mathbf{j} \) has components along the directions defined by these unit vectors.

To find the unit vector of any given vector \( \mathbf{A} \), you need to divide \( \mathbf{A} \) by its magnitude. The magnitude of \( \mathbf{A} = a\mathbf{i} + b\mathbf{j} \) is calculated as \( \|\mathbf{A}\| = \sqrt{a^2 + b^2} \). For instance, if \( \mathbf{W} = 3\mathbf{i} + 4\mathbf{j} \), its magnitude is \( \sqrt{3^2 + 4^2} = 5 \), so the unit vector \( \mathbf{w} = \frac{\mathbf{W}}{5} = \frac{3}{5}\mathbf{i} + \frac{4}{5}\mathbf{j} \). Unit vectors simplify the representation and calculations in physics and engineering, where knowing the direction is crucial.