Problem 57 - Precalculus with Calculus Previews

Question

In Problems \(49-62, \mathbf{u}=\langle 2,-3\rangle, \mathbf{v}=\langle-1,5\rangle,\) and \(\mathbf{w}=\langle 3,-2\rangle .\) Find the indicated scalar or vector. $$ (-\mathbf{v}) \cdot\left(\frac{1}{2} \mathbf{w}\right) $$

Step-by-Step Solution

Answer

The result of \\(-\mathbf{v} \cdot \left( \frac{1}{2} \mathbf{w} \right)\\) is \\(\frac{13}{2}\\).

Step 1: Find -v

To find \(-\mathbf{v}\), we need to take the negation of each component of vector \( \mathbf{v} = \langle -1, 5 \rangle\). Hence, \(-\mathbf{v} = \langle 1, -5 \rangle\).

Step 2: Compute \(\frac{1}{2} \mathbf{w}\)

The operation \( \frac{1}{2} \mathbf{w} \) involves multiplying each component of vector \( \mathbf{w} = \langle 3, -2 \rangle \) by \( \frac{1}{2} \). Thus, \( \frac{1}{2} \mathbf{w} = \left\langle \frac{3}{2}, -1 \right\rangle \).

Step 3: Calculate the Dot Product

Compute the dot product of \(-\mathbf{v} = \langle 1, -5 \rangle\) and \(\frac{1}{2} \mathbf{w} = \left\langle \frac{3}{2}, -1 \right\rangle\). The formula for the dot product is \( a \cdot b = a_1b_1 + a_2b_2 \). So, \( -\mathbf{v} \cdot \left( \frac{1}{2} \mathbf{w} \right) = 1 \times \frac{3}{2} + (-5) \times (-1) \).

Step 4: Solve the Expression

Calculate the values inside the dot product: \( 1 \times \frac{3}{2} = \frac{3}{2} \) and \((-5) \times (-1) = 5\). Then, sum these values: \( \frac{3}{2} + 5 = \frac{3}{2} + \frac{10}{2} = \frac{13}{2} \).

Key Concepts

Vector Negation

When working with vectors, vector negation involves changing the direction of a vector. This is achieved by reversing the sign of each component in the vector. For example, if we have a vector \( \mathbf{v} = \langle -1, 5 \rangle \), the negation of this vector, denoted as \( -\mathbf{v} \), results in \( \langle 1, -5 \rangle \).

This operation is useful when you need a vector pointing in the opposite direction of the original vector. It is a basic operation in vector mathematics and can be visualized geometrically as flipping the vector across the origin.

• **Original Vector:** \( \mathbf{v} = \langle -1, 5 \rangle \)
• **Negated Vector:** \( -\mathbf{v} = \langle 1, -5 \rangle \)

For our scenario, taking the negation of vector \( \mathbf{v} \) yields \( \langle 1, -5 \rangle \), which we then use in further calculations.

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying a vector by a scalar (a real number). This operation scales the vector by the given factor without altering its direction unless the scalar is negative.

To perform scalar multiplication, multiply each component of the vector by the scalar value. For vector \( \mathbf{w} = \langle 3, -2 \rangle \) and the scalar \( \frac{1}{2} \), the calculation is:

• **Multiply each component:**
• \( \frac{1}{2} \times 3 = \frac{3}{2} \)
• \( \frac{1}{2} \times -2 = -1 \)

This results in a new vector \( \langle \frac{3}{2}, -1 \rangle \). Scalar multiplication is essential in many vector operations, particularly when adjusting the length of a vector or performing linear transformations.

Dot Product Calculation

The dot product, also known as the scalar product, is a fundamental operation in vector mathematics. It takes two vectors and returns a single number (a scalar), providing insights into the vectors' relationship.

The formula for the dot product of two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \) is: \[ a \cdot b = a_1b_1 + a_2b_2 \] This operation can tell us whether vectors are orthogonal (dot product is zero), have the same direction, or opposite directions, among other things.

• **Example Calculation:** Find the dot product of \( -\mathbf{v} = \langle 1, -5 \rangle \) and \( \frac{1}{2} \mathbf{w} = \langle \frac{3}{2}, -1 \rangle \).
• Calculate: \( 1 \times \frac{3}{2} + (-5) \times (-1) \).
• Break it down: \( \frac{3}{2} + 5 = \frac{3}{2} + \frac{10}{2} = \frac{13}{2} \).

Understanding dot products is vital in physics, engineering, and computer graphics, where they are used to compute projections and measure angles between vectors.