Problem 25 - Contemporary Precalculus

Question

Find the components of the given vector, where \(\boldsymbol{u}=\boldsymbol{i}-2 \boldsymbol{j}, \boldsymbol{v}=3 \boldsymbol{i}+\boldsymbol{j}, \boldsymbol{w}=-4 \boldsymbol{i}+\boldsymbol{j}\) $$\frac{1}{4}(8 u+4 v-w)$$

Step-by-Step Solution

Answer

Answer: The components of the resulting vector are \((6, -\frac{13}{4})\).

Step 1: Find the components of the given vectors

The given vectors are: 1. \(\boldsymbol{u} = \boldsymbol{i} - 2 \boldsymbol{j}\), 2. \(\boldsymbol{v} = 3 \boldsymbol{i} + \boldsymbol{j}\), 3. \(\boldsymbol{w} = -4 \boldsymbol{i} + \boldsymbol{j}\) The components of \(\boldsymbol{u}\) are \((1, -2)\), the components of \(\boldsymbol{v}\) are \((3, 1)\), and the components of \(\boldsymbol{w}\) are \((-4, 1)\).

Step 2: Calculate the sum of the vectors

We need to calculate \(8u + 4v - w\). First, find the sum of the vectors by multiplying each vector by its corresponding scalar and adding them: $$8\boldsymbol{u} + 4\boldsymbol{v} - \boldsymbol{w} = (8(1, -2) + 4(3, 1) - (-4, 1))$$

Step 3: Perform the scalar multiplication

Multiply each vector by its scalar: $$8\boldsymbol{u} = 8(1, -2) = (8, -16)$$ $$4\boldsymbol{v} = 4(3, 1) = (12, 4)$$ Now, substitute these values back into the expression: $$(8, -16) + (12, 4) - (-4, 1)$$

Step 4: Perform the vector addition

Add the vectors: $$(8, -16) + (12, 4) = (20, -12)$$ $$(20, -12) - (-4, 1) = (24, -13)$$

Step 5: Calculate the final vector

Finally, multiply the resulting vector by \(\frac{1}{4}\): $$\frac{1}{4}(24, -13) = (6, -\frac{13}{4})$$ Therefore, the components of the given expression are \((6, -\frac{13}{4})\).

Key Concepts

Understanding Vector Addition

Vector addition is a fundamental operation in mathematics and physics. It's like combining forces or directions. When you add two vectors, you sum their respective components. This means:

\[\boldsymbol{a} = (a_1, a_2)\] and \[\boldsymbol{b} = (b_1, b_2)\] can be added as follows:

• First component: \(a_1 + b_1\)
• Second component: \(a_2 + b_2\)

So, the resulting vector \(\boldsymbol{c} = \boldsymbol{a} + \boldsymbol{b}\) is \((a_1 + b_1, a_2 + b_2)\).

In the solution, vector addition is used after scalar multiplication to handle the operations \(8\boldsymbol{u} + 4\boldsymbol{v} - \boldsymbol{w}\). Each operation transforms the original vectors, creating a new vector that represents the combined effect of all these vectors. This process is crucial for determining the final outcome of any vector problem.

Mastering Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a regular number). This operation stretches or shrinks the vector while maintaining its direction, unless the scalar is negative, which flips it.

Here's how it's done: If \(k\) is the scalar and \(\boldsymbol{a} = (a_1, a_2)\) is the vector:

• First component: \(k \times a_1\)
• Second component: \(k \times a_2\)

For example, multiplying \(\boldsymbol{u} = (1, -2)\) by 8 results in \((8 \times 1, 8 \times -2) = (8, -16)\).

In the exercise solution, scalar multiplication is performed before addition. This step ensures each vector is properly scaled according to its coefficient, resulting in correct calculations for vector addition.

Exploring Vector Components

Vectors are defined by their components, usually represented by the coordinates \((x, y)\) in a plane. These components denote the vector's position or direction in space. Understanding vector components helps in visualizing how vectors interact.

For the vectors in the problem:

• \(\boldsymbol{u} = (1, -2)\): It has a rightward movement of 1 and a downward movement of 2.
• \(\boldsymbol{v} = (3, 1)\): It moves 3 units right and 1 unit up.
• \(\boldsymbol{w} = (-4, 1)\): It moves 4 units left and 1 unit up.

Breaking vectors into components is essential. It allows complex operations to be simplified into addition and multiplication of numbers. The solution uses these components to compute the final result through a series of mathematical operations. Understanding each component's role is key to mastering vector mathematics.