Problem 69 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the component form of \(v\) and sketch the specified vector operations geometrically, where \(\mathbf{u}=2 \mathbf{i}-\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}+2 \mathbf{j}\).$$\mathbf{v}=\mathbf{u}+2 \mathbf{w}$$

Step-by-Step Solution

Answer

The component form of the vector \( \mathbf{v} \) is \( 4\mathbf{i} + 3\mathbf{j} \)

Step 1: Understand Given Vectors

The given vectors are \( \mathbf{u} = 2\mathbf{i} - \mathbf{j} \) and \( \mathbf{w} = \mathbf{i} + 2 \mathbf{j} \). These vectors are already in component form.

Step 2: Perform Scalar Multiplication

The exercise asks for \( \mathbf{v} = \mathbf{u} + 2\mathbf{w} \), so first, multiply the vector \( \mathbf{w} \) by the scalar 2. This results in \( 2\mathbf{w} = 2(\mathbf{i}) + 4(\mathbf{j}) = 2\mathbf{i} + 4\mathbf{j} \).

Step 3: Implement Vector Addition

Now add vector \( \mathbf{u} \) to \( 2\mathbf{w} \) as stated in the equation \( \mathbf{v} = \mathbf{u} + 2\mathbf{w} \). This implies \( \mathbf{v} = (2\mathbf{i} - \mathbf{j}) + (2\mathbf{i} + 4\mathbf{j}) = 4\mathbf{i} + 3\mathbf{j} \).

Step 4: Write Final Answer

The component form of the vector \( \mathbf{v} \) is \( 4\mathbf{i} + 3\mathbf{j} \).

Key Concepts

Understanding Component Form

Vectors are often represented in component form, making it easier to see their impact in a given space. When a vector such as \( \mathbf{u} = 2\mathbf{i} - \mathbf{j} \) is expressed in component form, the components \( 2 \) and \( -1 \) indicate its influence in the horizontal \( i \) and vertical \( j \) directions, respectively.

This format is particularly helpful in visualizing and performing further operations, as it breaks the vector down into its individual axis components. These components can then be manipulated by operations like scalar multiplication or vector addition.

In the given exercise, both vectors \( \mathbf{u} \) and \( \mathbf{w} \) are already in component form, simplifying the calculations and enabling us to directly see the effects of each operation.

Exploring Scalar Multiplication

Scalar multiplication is a fundamental vector operation that involves multiplying a vector by a number, known as a scalar. This changes the vector's length while keeping it in the same direction if the scalar is positive, or reversing its direction if the scalar is negative.

In our example, we need to multiply vector \( \mathbf{w} = \mathbf{i} + 2 \mathbf{j} \) by the scalar \( 2 \). We achieve this by multiplying each component of \( \mathbf{w} \) by \( 2 \):

• This changes the \( i \)-component from \( 1 \) to \( 2 \times 1 = 2 \).
• Similarly, the \( j \)-component goes from \( 2 \) to \( 2 \times 2 = 4 \).

Therefore, after scalar multiplication, vector \( 2 \mathbf{w} = 2\mathbf{i} + 4\mathbf{j} \). This step demonstrates how scalar multiplication scales a vector to affect its influence in space, without altering its original direction.

Performing Vector Addition

Vector addition is another crucial vector operation where two or more vectors are combined to form a single resultant vector. This operation involves adding each pair of corresponding components.

For the exercise, we compute \( \mathbf{v} = \mathbf{u} + 2 \mathbf{w} \), requiring the addition of \( \mathbf{u} = 2\mathbf{i} - \mathbf{j} \) and \( 2\mathbf{w} = 2\mathbf{i} + 4\mathbf{j} \).

To add these vectors, we sum the \( i \)-components and the \( j \)-components separately:

• The \( i \)-components: \( 2 + 2 = 4 \).
• The \( j \)-components: \( -1 + 4 = 3 \).

This results in the vector \( \mathbf{v} = 4\mathbf{i} + 3\mathbf{j} \). Vector addition helps to easily determine the effect of combining multiple vectors, illustrating a cumulative impact along the axes.