Problem 19 - Precalculus with Calculus Previews

Question

Find \(\mathbf{u}-4 \mathbf{v}\) and \(2 \mathbf{u}+5 \mathbf{v}\). $$ \mathbf{u}=0.2 \mathbf{i}+0.1 \mathbf{j}, \mathbf{v}=-1.4 \mathbf{i}-2.1 \mathbf{j} $$

Step-by-Step Solution

Answer

\( \mathbf{u} - 4\mathbf{v} = 5.8 \mathbf{i} + 8.5 \mathbf{j} \), \( 2\mathbf{u} + 5\mathbf{v} = -6.6 \mathbf{i} - 10.3 \mathbf{j} \).

Step 1: Understand the Vectors

First, identify the given vectors: \( \mathbf{u} = 0.2 \mathbf{i} + 0.1 \mathbf{j} \) and \( \mathbf{v} = -1.4 \mathbf{i} - 2.1 \mathbf{j} \). We need to find two results: \( \mathbf{u} - 4\mathbf{v} \) and \( 2\mathbf{u} + 5\mathbf{v} \).

Step 2: Multiply Vector \(\mathbf{v}\) by 4

Calculate \( 4\mathbf{v} \) by multiplying each component of \( \mathbf{v} \) by 4. \[ 4\mathbf{v} = 4(-1.4 \mathbf{i} - 2.1 \mathbf{j}) = (-5.6 \mathbf{i} - 8.4 \mathbf{j}) \]

Step 3: Subtract \( 4\mathbf{v} \) from \( \mathbf{u} \)

Now, subtract \( 4\mathbf{v} \) from \( \mathbf{u} \) by subtracting the corresponding components.\[ \mathbf{u} - 4\mathbf{v} = (0.2 \mathbf{i} + 0.1 \mathbf{j}) - (-5.6 \mathbf{i} - 8.4 \mathbf{j}) \]Calculate:\[ = (0.2 + 5.6 ) \mathbf{i} + (0.1 + 8.4) \mathbf{j} \]\[ = 5.8 \mathbf{i} + 8.5 \mathbf{j} \]

Step 4: Multiply Vector \(\mathbf{u}\) by 2

Calculate \( 2\mathbf{u} \) by multiplying each component of \( \mathbf{u} \) by 2.\[ 2\mathbf{u} = 2(0.2 \mathbf{i} + 0.1 \mathbf{j}) = (0.4 \mathbf{i} + 0.2 \mathbf{j}) \]

Step 5: Multiply Vector \(\mathbf{v}\) by 5

Calculate \( 5\mathbf{v} \) by multiplying each component of \( \mathbf{v} \) by 5.\[ 5\mathbf{v} = 5(-1.4 \mathbf{i} - 2.1 \mathbf{j}) = (-7.0 \mathbf{i} - 10.5 \mathbf{j}) \]

Step 6: Add \( 2\mathbf{u} \) and \( 5\mathbf{v} \)

Combine the results from Steps 4 and 5 by adding the corresponding components.\[ 2\mathbf{u} + 5\mathbf{v} = (0.4 \mathbf{i} + 0.2 \mathbf{j}) + (-7.0 \mathbf{i} - 10.5 \mathbf{j}) \]Calculate: \[ = (0.4 - 7.0) \mathbf{i} + (0.2 - 10.5) \mathbf{j} \]\[ = -6.6 \mathbf{i} - 10.3 \mathbf{j} \]

Key Concepts

Vector Addition

Adding vectors involves combining their respective components. You can think of it as putting together their effects or displacements. For example, to find the result of combining two vectors, like in the step where we calculated \(2\mathbf{u} + 5\mathbf{v}\), you should:

• Add their i-components together.
• Add their j-components together.

This process creates a new vector, often referred to as the resultant vector, which represents the total effect of the two vectors. In short, if you have vector \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and vector \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \), the sum \( \mathbf{a} + \mathbf{b} \) is calculated by \((a_1 + b_1) \mathbf{i} + (a_2 + b_2) \mathbf{j} \). Think of it as walking two paths: the first path determined by \( \mathbf{a} \), and the second path determined by \( \mathbf{b} \). The resultant vector shows where you'd end up if you walked both paths in sequence.

Scalar Multiplication

Scalar multiplication of vectors is a key concept in vector algebra and involves changing the 'length' of a vector. We multiply each component of the vector by the scalar. For example, when the problem asks for \(4\mathbf{v}\), it means multiplying \(\mathbf{v}\) by 4:

• Multiply the i-component by 4.
• Multiply the j-component by 4.

This operation scales the vector. If the scalar is greater than 1, the vector lengthens (in the same direction). If the scalar is between 0 and 1, the vector shortens. If the scalar is negative, it not only changes the length but also reverses the direction. For instance, multiplying vector \( \mathbf{v} = -1.4 \mathbf{i} - 2.1 \mathbf{j} \) by 4 gives \(-5.6 \mathbf{i} - 8.4 \mathbf{j} \). This result represents a vector that is four times longer and in the same direction as \( \mathbf{v} \).

Vector Subtraction

Vector subtraction is akin to adding a negative. When you subtract vectors, such as finding \(\mathbf{u} - 4\mathbf{v}\), you need to

• Subtract the i-component of the second vector from the i-component of the first vector.
• Subtract the j-component of the second vector from the j-component of the first vector.

In essence, vector subtraction can be transformed into an addition problem by thinking of the subtraction as adding the opposite vector. For example, subtracting vector \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \) from vector \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) is the same as adding \( -b_1 \mathbf{i} - b_2 \mathbf{j} \). In our specific example, this operation resulted in \( 5.8 \mathbf{i} + 8.5 \mathbf{j} \) as the subtraction of \( 4\mathbf{v} \) from \( \mathbf{u} \). This highlights that where one vector 'ends', the other 'begins', showing the removal of \(4\mathbf{v}\)'s effect from \(\mathbf{u}\).