Problem 26 - Contemporary Precalculus

Question

Find the components of the given vector, where \(\boldsymbol{u}=\boldsymbol{i}-2 \boldsymbol{j}, \boldsymbol{v}=3 \boldsymbol{i}+\boldsymbol{j}, \boldsymbol{w}=-4 \boldsymbol{i}+\boldsymbol{j}\) $$3(\mathbf{u}-2 \mathbf{v})-6 \mathbf{w}$$

Step-by-Step Solution

Answer

Question: Calculate the components of the given vector expression: \(3(\boldsymbol{u} - 2\boldsymbol{v}) - 6\boldsymbol{w}\) where \(\boldsymbol{u} = \boldsymbol{i} - 2\boldsymbol{j}\), \(\boldsymbol{v} = 3\boldsymbol{i} + \boldsymbol{j}\), and \(\boldsymbol{w} = -4\boldsymbol{i} + \boldsymbol{j}\). Answer: The components of the given vector expression are \(9\boldsymbol{i}-18\boldsymbol{j}\).

Step 1: Perform the subtraction and scalar multiplication inside the parentheses

First, perform the operation inside the parentheses. Subtract \(2\boldsymbol{v}\) from \(\boldsymbol{u}\) and then multiply the result by the scalar \(3\): $$3(\boldsymbol{u} - 2\boldsymbol{v}) = 3((\boldsymbol{i}-2 \boldsymbol{j}) - 2(3\boldsymbol{i}+\boldsymbol{j}))$$

Step 2: Calculate the subtraction and scalar multiplication

Now we need to calculate the subtraction and multiplication: $$3(\boldsymbol{i}-2 \boldsymbol{j}) - 6(3\boldsymbol{i}+\boldsymbol{j}) = 3\boldsymbol{i} - 6\boldsymbol{j} - 18\boldsymbol{i} - 6\boldsymbol{j}$$

Step 3: Perform the addition of the vectors

Combine the like terms (the i components and the j components): $$3\boldsymbol{i} - 18\boldsymbol{i} - 6\boldsymbol{j} - 6\boldsymbol{j} = (-15\boldsymbol{i}-12\boldsymbol{j})$$

Step 4: Perform the subtraction with the third vector

Now, subtract the result from Step 3 by \(6\boldsymbol{w}\): $$(-15\boldsymbol{i}-12\boldsymbol{j}) - 6(-4\boldsymbol{i}+\boldsymbol{j})$$

Step 5: Calculate the subtraction and scalar multiplication

Next, carry out the subtraction and scalar multiplication: $$(-15\boldsymbol{i}-12\boldsymbol{j}) - 6(-4\boldsymbol{i}+\boldsymbol{j}) = -15\boldsymbol{i} - 12\boldsymbol{j} + 24\boldsymbol{i} - 6\boldsymbol{j}$$

Step 6: Add the resultant vectors

Finally, add the vectors from Step 5 to get the final resultant vector: $$-15\boldsymbol{i} + 24\boldsymbol{i} - 12\boldsymbol{j} - 6\boldsymbol{j} = (9\boldsymbol{i}-18\boldsymbol{j})$$ The components of the given vector expression are \(9\boldsymbol{i}-18\boldsymbol{j}\).

Key Concepts

vector subtraction

In vector mathematics, subtraction is similar to traditional subtraction, but we work with each component separately. To subtract one vector from another, we subtract their respective components. For example, if we have vectors \( \boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} \) and \( \boldsymbol{b} = b_1 \boldsymbol{i} + b_2 \boldsymbol{j} \), the difference \( \boldsymbol{a} - \boldsymbol{b} \) becomes:

• Subtract the i-components: \( (a_1 - b_1) \boldsymbol{i} \)
• Subtract the j-components: \( (a_2 - b_2) \boldsymbol{j} \)

In the original exercise, this operation was used to subtract \( 2\boldsymbol{v} \) from \( \boldsymbol{u} \). Each component of \( 2\boldsymbol{v} \) was subtracted from the components of \( \boldsymbol{u} \) to yield a new vector.

scalar multiplication

Scalar multiplication involves multiplying a vector by a scalar (a real number), which scales the vector's magnitude without changing its direction. When you multiply a vector \( \boldsymbol{v} = v_1 \boldsymbol{i} + v_2 \boldsymbol{j} \) by a scalar \( k \), the resultant vector is simply:

• Each component is multiplied by the scalar: \( (k \cdot v_1) \boldsymbol{i} + (k \cdot v_2) \boldsymbol{j} \).

For example, in the exercise, the expression \( 3(\boldsymbol{u} - 2\boldsymbol{v}) \) involves multiplying each component of the vector resulting from subtraction by the scalar \( 3 \). This scaling step simplifies the vector and is essential for calculating the final result.

vector components

A vector in a 2-dimensional space is represented using two components, usually in relation to the unit vectors \( \boldsymbol{i} \) and \( \boldsymbol{j} \). These components describe the vector's position parallel to the x-axis and y-axis.

• The \( i \)-component indicates the vector's horizontal contribution and aligns with the x-axis.
• The \( j \)-component represents the vector's vertical aspect and aligns with the y-axis.

In the original exercise, the vectors \( \boldsymbol{u} \), \( \boldsymbol{v} \), and \( \boldsymbol{w} \) are expressed with their respective components, enabling operations like addition and subtraction to be carried out easily one axis at a time.

vector addition

Vector addition is the process of combining two or more vectors to find a resultant vector. This is done component-wise, which means you add the corresponding components of each vector. Consider vectors \( \boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} \) and \( \boldsymbol{b} = b_1 \boldsymbol{i} + b_2 \boldsymbol{j} \).

• Add the i-components: \( (a_1 + b_1) \boldsymbol{i} \)
• Add the j-components: \( (a_2 + b_2) \boldsymbol{j} \)

In the provided exercise, vector addition was employed in the final steps to combine various results from prior operations. This process leads to obtaining the final vector, expressed with new combined components, which simplifies down to the final expression \( 9\boldsymbol{i} - 18\boldsymbol{j} \).