Problem 36 - Algebra and Trigonometry

Question

\(31-36\) Find \(2 \mathbf{u},-3 \mathbf{v}, \mathbf{u}+\mathbf{v},\) and \(3 \mathbf{u}-4 \mathbf{v}\) for the given vectors \(\mathbf{u}\) and \(\mathbf{v} .\) $$ \mathbf{u}=\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=\mathbf{i}-\mathbf{j} $$

Step-by-Step Solution

Answer

\(2\mathbf{u}=2\mathbf{i}+2\mathbf{j}\), \(-3\mathbf{v}=-3\mathbf{i}+3\mathbf{j}\), \(\mathbf{u}+\mathbf{v}=2\mathbf{i}\), \(3\mathbf{u}-4\mathbf{v}=7\mathbf{i}-\mathbf{j}\).

Step 1: Calculate \(2 \mathbf{u}\)

First, multiply vector \(\mathbf{u}\) by 2. Given \(\mathbf{u} = \mathbf{i} + \mathbf{j}\), multiply each component by 2: \[ 2 \mathbf{u} = 2(\mathbf{i} + \mathbf{j}) = 2\mathbf{i} + 2\mathbf{j} \]

Step 2: Calculate \(-3 \mathbf{v}\)

Now, multiply vector \(\mathbf{v}\) by \(-3\). Given \(\mathbf{v} = \mathbf{i} - \mathbf{j}\), multiply each component by \(-3\): \[ -3 \mathbf{v} = -3(\mathbf{i} - \mathbf{j}) = -3\mathbf{i} + 3\mathbf{j} \]

Step 3: Calculate \(\mathbf{u} + \mathbf{v}\)

Add vectors \(\mathbf{u}\) and \(\mathbf{v}\) together. Using \(\mathbf{u} = \mathbf{i} + \mathbf{j}\) and \(\mathbf{v} = \mathbf{i} - \mathbf{j}\): \[ \mathbf{u} + \mathbf{v} = (\mathbf{i} + \mathbf{j}) + (\mathbf{i} - \mathbf{j}) = 2\mathbf{i} + 0\mathbf{j} = 2\mathbf{i} \]

Step 4: Calculate \(3 \mathbf{u} - 4 \mathbf{v}\)

Combine vectors \(3 \mathbf{u}\) and \(-4 \mathbf{v}\). Calculate each separately first:\[ 3 \mathbf{u} = 3(\mathbf{i} + \mathbf{j}) = 3\mathbf{i} + 3\mathbf{j} \]\[ -4 \mathbf{v} = -4(\mathbf{i} - \mathbf{j}) = -4\mathbf{i} + 4\mathbf{j} \]Now subtract \(4 \mathbf{v}\) from \(3 \mathbf{u}\):\[ 3\mathbf{u} - 4\mathbf{v} = (3\mathbf{i} + 3\mathbf{j}) - (-4\mathbf{i} + 4\mathbf{j}) = (3 + 4)\mathbf{i} + (3 - 4)\mathbf{j} = 7\mathbf{i} - \mathbf{j} \]

Key Concepts

Scalar Multiplication of Vectors

Scalar multiplication is a fundamental operation in vector algebra. It involves multiplying a vector by a scalar (a real number), which results in a new vector. This operation affects the length of the vector, while maintaining its direction if the scalar is positive, or reversing its direction if the scalar is negative.

Consider vector \( \mathbf{u} = \mathbf{i} + \mathbf{j} \). If we multiply \( \mathbf{u} \) by a scalar 2, we perform the operation on each component of the vector separately:

• Multiply the \( \mathbf{i} \) component by 2, resulting in \( 2\mathbf{i} \).
• Multiply the \( \mathbf{j} \) component by 2, resulting in \( 2\mathbf{j} \).

Therefore, the result of \( 2 \mathbf{u} \) is \( 2\mathbf{i} + 2\mathbf{j} \).

Similarly, with vector \( \mathbf{v} = \mathbf{i} - \mathbf{j} \), multiplying by \(-3\) gives us:

• \(-3 \mathbf{i} \) for the \( \mathbf{i} \) component.
• \(+3 \mathbf{j} \) for the \( \mathbf{j} \) component (because multiplying a negative by a negative is positive).

Thus, \(-3 \mathbf{v} = -3\mathbf{i} + 3\mathbf{j} \).

Vector Addition

Vector addition involves adding corresponding components of vectors to result in a new vector. This operation is straightforward and visually represents the combination of two vectors by joining them head-to-tail.

For vectors \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{v} = \mathbf{i} - \mathbf{j} \), vector addition would look like this:

• Add \( \mathbf{i} \) components: \( \mathbf{i} + \mathbf{i} = 2\mathbf{i} \).
• Add \( \mathbf{j} \) components: \( \mathbf{j} - \mathbf{j} = 0 \mathbf{j} \).

Thus, \( \mathbf{u} + \mathbf{v} = 2\mathbf{i} \).

This operation demonstrates how vector components are directly combined to form a resultant vector. The simplified result \( 2\mathbf{i} \) indicates a vector purely in the direction of the positive \( \mathbf{i} \)-axis, showing how components directly oppose each other can cancel out.

Vector Subtraction

Vector subtraction can be thought of as adding a negative vector. This method is used when you need to find the difference between two vectors. In math terms, it's equivalent to adding the opposite of the vector you want to subtract from the first vector.

When you evaluate the expression \( 3\mathbf{u} - 4\mathbf{v} \):

• First, calculate \( 3\mathbf{u} \) by scalar multiplication: \( 3(\mathbf{i} + \mathbf{j}) = 3\mathbf{i} + 3\mathbf{j} \).
• Then, calculate \(-4\mathbf{v} \) as described earlier: \(-4(\mathbf{i} - \mathbf{j}) = -4\mathbf{i} + 4\mathbf{j} \).

Then combine them with subtraction:

• \( (3\mathbf{i} + 3\mathbf{j}) - (-4\mathbf{i} + 4\mathbf{j}) \).
• This becomes \( (3 + 4)\mathbf{i} + (3 - 4)\mathbf{j} \).
• Resulting in \( 7\mathbf{i} - \mathbf{j} \).

The final vector \( 7\mathbf{i} - \mathbf{j} \) illustrates how subtracting vectors can be used to produce a new vector by combining individual components using basic arithmetic.