Problem 21 - Algebra and Trigonometry

Question

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 \(\mathbf{v} .\) $$ \mathbf{u}=2 \mathbf{i}, \quad \mathbf{v}=3 \mathbf{i}-2 \mathbf{j} $$

Step-by-Step Solution

Answer

\(2 \mathbf{u} = 4 \mathbf{i}, -3 \mathbf{v} = -9 \mathbf{i} + 6 \mathbf{j}, \mathbf{u} + \mathbf{v} = 5 \mathbf{i} - 2 \mathbf{j}, 3 \mathbf{u} - 4 \mathbf{v} = 18 \mathbf{i} - 8 \mathbf{j}\).

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

Multiply each component of \(\mathbf{u}\) by 2. Since \(\mathbf{u}=2 \mathbf{i}\), we have \(2 \mathbf{u} = 2 \times 2 \mathbf{i} = 4 \mathbf{i}\).

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

Multiply each component of \(\mathbf{v}\) by -3. Given \(\mathbf{v}=3 \mathbf{i}-2 \mathbf{j}\), calculate:\(-3 \mathbf{v} = -3 \times (3 \mathbf{i} - 2 \mathbf{j}) = -9 \mathbf{i} + 6 \mathbf{j}\).

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

Add the corresponding components of \(\mathbf{u}\) and \(\mathbf{v}\). With \(\mathbf{u} = 2\mathbf{i}\) and \(\mathbf{v} = 3\mathbf{i} - 2\mathbf{j}\), find:\(\mathbf{u} + \mathbf{v} = 2\mathbf{i} + (3\mathbf{i} - 2\mathbf{j}) = (2+3)\mathbf{i} - 2 \mathbf{j} = 5\mathbf{i} - 2\mathbf{j}\).

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

First, find \(3 \mathbf{u}\) by multiplying each component of \(\mathbf{u}\) by 3: \(3 \mathbf{u} = 3 \times 2 \mathbf{i} = 6 \mathbf{i}\).Now, find \(-4 \mathbf{v}\) by multiplying each component of \(\mathbf{v}\) by -4: \(-4 \mathbf{v} = -4 \times (3\mathbf{i} - 2\mathbf{j}) = -12 \mathbf{i} + 8 \mathbf{j}\). Finally, subtract \(-4 \mathbf{v}\) from \(3 \mathbf{u}\):\(3 \mathbf{u} - 4 \mathbf{v} = 6 \mathbf{i} - (-12 \mathbf{i} + 8 \mathbf{j}) = (6 + 12) \mathbf{i} - 8 \mathbf{j} = 18 \mathbf{i} - 8 \mathbf{j}\).

Key Concepts

Scalar Multiplication

Scalar multiplication involves taking a vector and multiplying it by a scalar (a real number). Imagine you have a vector pointing in a certain direction, with a certain length. When you multiply the vector by a scalar, you're essentially stretching or compressing it. In some cases, this can also involve reversing the direction if the scalar is negative.

For example, consider the vector \( \mathbf{u} = 2 \mathbf{i} \). If we multiply by the scalar 2, we perform the calculation \( 2 \times 2 \mathbf{i} = 4 \mathbf{i} \). This results in a new vector still pointing in the original direction but now twice as long.

Similarly, when working with the vector \( \mathbf{v} = 3 \mathbf{i} - 2 \mathbf{j} \), applying a scalar \(-3\) involves:\(-3 \mathbf{v} = -3 \times (3 \mathbf{i} - 2 \mathbf{j}) = -9 \mathbf{i} + 6 \mathbf{j} \). This operation resulted in reversing the vector while stretching it threefold. Think of scalar multiplication as a simple arithmetic operation that might change both the length and direction of a vector.

Vector Addition

Vector addition is akin to combining two arrows in a plane. Each arrow's tail meets the other's head to form a new resultant vector. This process is essential in fields like physics, where combining different forces or velocities is a common task.

When adding vectors, you add corresponding components together. Take the vectors \( \mathbf{u} = 2\mathbf{i} \) and \( \mathbf{v} = 3\mathbf{i} - 2\mathbf{j} \). Adding these gives:

• First, add the \(i\)-components: \(2 + 3 = 5\).
• Then, add the \(j\)-components: there's only \(-2 \mathbf{j}\) from \( \mathbf{v} \).

Thus, the outcome of \( \mathbf{u} + \mathbf{v} \) is \( 5 \mathbf{i} - 2 \mathbf{j} \). It's like placing arrows tip-to-tail to find their total effect.

This method ensures that your addition respects the direction and magnitude of each vector, making it a reliable way to understand combined movements or forces in multi-dimensional space.

Vector Subtraction

Vector subtraction is similar to addition but with an important twist: instead of combining vectors, you're finding out the difference between them. Imagine you're trying to reverse the effect of one vector from another. The goal is to calculate the additional direction and length required to balance one vector with the other.

Subtracting vectors means you are essentially adding a negative version of the vector. Consider the example of calculating \(3 \mathbf{u} - 4 \mathbf{v}\). Start by multiplying each vector by the necessary scalar:

• \(3 \mathbf{u} = 6 \mathbf{i}\)
• \(-4 \mathbf{v} = -12 \mathbf{i} + 8 \mathbf{j}\)

Now, perform the subtraction operation, which is the same as addition throughout by treating the negative sign:
\[3 \mathbf{u} - 4 \mathbf{v} = 6 \mathbf{i} - (-12 \mathbf{i} + 8 \mathbf{j})\].
This simplifies to \(18 \mathbf{i} - 8 \mathbf{j}\).

Vector subtraction gives you the resultant vector needed to "cancel out" the effects of the vector being subtracted, an essential operation for balancing forces or finding opposites in vector space.