Problem 30 - Precalculus with Calculus Previews

Question

If \(\mathbf{u}=3 \mathbf{i}-\mathbf{j}\) and \(\mathbf{v}=2 \mathbf{i}+4 \mathbf{j},\) find the horizontal and the vertical components of the indicated vector. $$ 3(\mathbf{u}+\mathbf{v}) $$

Step-by-Step Solution

Answer

The horizontal component is 15, and the vertical component is 9.

Step 1: Express the Vectors

Given the vectors \( \mathbf{u} = 3 \mathbf{i} - \mathbf{j} \) and \( \mathbf{v} = 2 \mathbf{i} + 4 \mathbf{j} \), let's write them explicitly as \( \mathbf{u} = \begin{pmatrix} 3 \ -1 \end{pmatrix} \) and \( \mathbf{v} = \begin{pmatrix} 2 \ 4 \end{pmatrix} \).

Step 2: Add the Vectors

Add the vectors \( \mathbf{u} \) and \( \mathbf{v} \) component-wise. This means adding their horizontal components (\( 3 + 2 \)) and their vertical components (\( -1 + 4 \)). So, \( \mathbf{u} + \mathbf{v} = \begin{pmatrix} 3+2 \ -1+4 \end{pmatrix} = \begin{pmatrix} 5 \ 3 \end{pmatrix} \).

Step 3: Scale the Resulting Vector

Multiply the resulting vector \( \begin{pmatrix} 5 \ 3 \end{pmatrix} \) by 3 to get \( 3(\mathbf{u} + \mathbf{v}) = 3 \times \begin{pmatrix} 5 \ 3 \end{pmatrix} = \begin{pmatrix} 3 \times 5 \ 3 \times 3 \end{pmatrix} = \begin{pmatrix} 15 \ 9 \end{pmatrix} \).

Step 4: Identify Horizontal and Vertical Components

The resulting vector \( \begin{pmatrix} 15 \ 9 \end{pmatrix} \) represents the vector \( 3(\mathbf{u} + \mathbf{v}) \). The horizontal component is 15, and the vertical component is 9.

Key Concepts

Vector Components

Vectors are mathematical entities that have both a magnitude and a direction. In a 2-dimensional space, vectors can be broken down into two components: a horizontal component and a vertical component. Think of these components as the legs of a right triangle where the vector is the hypotenuse.

• Horizontal Component: Determines how far, or towards the left or right, a vector points.
• Vertical Component: Indicates how far up or down a vector points.

These components are usually represented using unit vectors, which are \( \mathbf{i} \) for the horizontal direction and \( \mathbf{j} \) for the vertical direction. When a vector \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) is given, \( a_1 \) and \( a_2 \) are the magnitudes of the horizontal and vertical components respectively.

Scalar Multiplication

Scalar multiplication involves changing the size of a vector by multiplying it with a scalar (a real number). This operation affects both the magnitude and, potentially, the direction of the vector:

• If the scalar is positive, the direction of the vector remains unchanged.
• If the scalar is negative, the direction will be reversed.

The effect of scalar multiplication is that each component of the vector is multiplied by the scalar. For example, if we have a vector \( \mathbf{a} = \begin{pmatrix} x \, y \end{pmatrix} \) and a scalar \( c \), then the resulting vector after scalar multiplication becomes \( \begin{pmatrix} c \cdot x \, c \cdot y \end{pmatrix} \). This operation is crucial when scaling vectors like in the exercise, where a vector has been multiplied by 3. This scaling increases both the horizontal and vertical components proportionally.

Horizontal Component

The horizontal component of a vector is an essential part of determining the vector's full effect, especially in physical systems. To find it, you simply read or calculate the coefficient of the \( \mathbf{i} \)-component in your vector expression.
For example, if a vector \( \mathbf{a} = 6 \mathbf{i} + 2 \mathbf{j} \) is given, the horizontal component is 6.
In our specific exercise:

• We first added the horizontal components of vectors \( \mathbf{u} = 3 \mathbf{i} - \mathbf{j} \) and \( \mathbf{v} = 2 \mathbf{i} + 4 \mathbf{j} \), giving us \( 3 + 2 = 5 \).
• Then, after scaling by 3, our horizontal component became \( 3 \times 5 = 15 \).

This final value of 15 represents how much the resulting vector points in the positive horizontal direction after performing vector addition and scalar multiplication.

Vertical Component

The vertical component measures how much a vector points upward or downward. It's determined by reading or calculating the coefficient of the \( \mathbf{j} \)-component.
In a vector like \( \mathbf{b} = 3 \mathbf{i} + 7 \mathbf{j} \), the vertical component is 7.
By using our exercise:

• You first add the vertical components of vectors \( \mathbf{u} = 3 \mathbf{i} - \mathbf{j} \) and \( \mathbf{v} = 2 \mathbf{i} + 4 \mathbf{j} \), giving \( -1 + 4 = 3 \).
• Then multiply by 3 to scale, resulting in a vertical component of \( 3 \times 3 = 9 \).

This vertical component indicates the extent of the vector's motion in the upward direction after all operations, much like how the horizontal component described lateral motion. Together, these components fully describe the scaled vector.