Problem 31 - 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. $$ \mathbf{v}-4 \mathbf{u} $$

Step-by-Step Solution

Answer

Horizontal component is 14; vertical component is 0.

Step 1: Understanding the Vectors

First, we need to understand the given vectors \(\mathbf{u}\) and \(\mathbf{v}\). The vector \(\mathbf{u}\) is represented as \(3\mathbf{i} - \mathbf{j}\), which means its horizontal component (along \(\mathbf{i}\)) is 3 and its vertical component (along \(\mathbf{j}\)) is -1. The vector \(\mathbf{v}\) is given as \(2\mathbf{i} + 4\mathbf{j}\), with a horizontal component of 2 and a vertical component of 4.

Step 2: Apply Scalar Multiplication

We need to find \(-4\mathbf{u}\). To do this, multiply both the horizontal and vertical components of \(\mathbf{u}\) by -4: \(-4\cdot (3\mathbf{i} - \mathbf{j}) = -4\times 3\mathbf{i} - 4\times(-1)\mathbf{j} = -12\mathbf{i} + 4\mathbf{j}\).

Step 3: Subtract Vectors

Now, subtract the vector \(-4\mathbf{u}\) from \(\mathbf{v}\): \(\mathbf{v} - 4\mathbf{u} = (2\mathbf{i} + 4\mathbf{j}) - (-12\mathbf{i} + 4\mathbf{j})\).

Step 4: Combine Like Terms

Combine the like terms from the vectors to find the resultant vector: \((2\mathbf{i} - (-12\mathbf{i})) + (4\mathbf{j} - 4\mathbf{j}) = (2 + 12)\mathbf{i} + (4 - 4)\mathbf{j} = 14\mathbf{i} + 0\mathbf{j}\).

Step 5: Identify Components

The resulting vector \(14\mathbf{i} + 0\mathbf{j}\) has a horizontal component of 14 and a vertical component of 0, meaning all the vector's direction and magnitude lie along the horizontal axis.

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation when dealing with vectors. It involves multiplying a vector by a scalar (a simple number), which affects both the direction and the magnitude of the vector. When you multiply a vector by a scalar, each component of the vector is multiplied by that scalar.

For the vector \( \mathbf{u} = 3 \mathbf{i} - \mathbf{j} \), and a scalar value of \( -4 \), scalar multiplication involves multiplying each component by \( -4 \):

- Horizontal component: \( -4 \times 3 = -12 \)
- Vertical component: \( -4 \times (-1) = 4 \)

This results in a new vector \( -12\mathbf{i} + 4\mathbf{j} \).

Scalar multiplication can reverse the direction of the vector if the scalar is negative, and it stretches or shrinks the vector depending on the absolute value of the scalar.

Vector Subtraction

Vector subtraction is the process of finding the difference between two vectors. It involves subtracting corresponding components from one vector to another. A simple way to think of this operation is like finding the resultant of applying one vector and then reversing the second vector.

For example, if you have two vectors \( \mathbf{v} = 2\mathbf{i} + 4\mathbf{j} \) and \( -4\mathbf{u} = -12\mathbf{i} + 4\mathbf{j} \), subtract \( -4\mathbf{u} \) from \( \mathbf{v} \) as follows:

- Subtract the horizontal components: \( 2 - (-12) = 2 + 12 = 14 \)
- Subtract the vertical components: \( 4 - 4 = 0 \)

This results in the vector \( 14\mathbf{i} + 0\mathbf{j} \).

Vector subtraction is crucial because it helps in understanding relative motion and finding how far or in what direction one vector is from another.

Horizontal and Vertical Components

Every vector can be broken down into horizontal and vertical components, which represent the vector’s influence in the horizontal and vertical directions respectively. This breakdown is helpful in solving real-world problems involving forces, movement, and direction.

For any vector \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} \), the horizontal component is \( a_1 \), and the vertical component is \( a_2 \). In our example with the vector \( 14\mathbf{i} + 0\mathbf{j} \), the components are:

• Horizontal: 14
• Vertical: 0

These components indicate that all projection occurs along the horizontal axis and there is no vertical movement or force.

Understanding and calculating these components provides insight into the vector's direction and effectiveness in both horizontal and vertical alignments.