Problem 78 - A Graphical Approach to Precalculus with Limits

Question

Let \(\mathbf{u}=\langle- 2,1\rangle, \mathbf{v}=\langle 3,4\rangle,\) and \(\mathbf{w}=\langle- 5,12\rangle .\) Evaluate each expression. $$\mathbf{u} \cdot(3 \mathbf{v})$$

Step-by-Step Solution

Answer

The value of \(\mathbf{u} \cdot (3\mathbf{v})\) is \(-6\).

Step 1: Calculate 3 Times Vector v

To find \(3\mathbf{v}\), multiply each component of \(\mathbf{v} = \langle 3, 4 \rangle\) by 3. This gives: \(3 \times 3 = 9\) and \(3 \times 4 = 12\), resulting in \(3\mathbf{v} = \langle 9, 12 \rangle\).

Step 2: Dot Product of u and 3v

Calculate the dot product \(\mathbf{u} \cdot (3\mathbf{v})\) using \(\mathbf{u} = \langle -2, 1 \rangle\) and \(3\mathbf{v} = \langle 9, 12 \rangle\). The formula for dot product is \(a_1b_1 + a_2b_2\). For our vectors, this is:\((-2) \times 9 + 1 \times 12 = -18 + 12 = -6\).

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a crucial operation in vector algebra. When working with two vectors, it helps find a single scalar value. This operation combines corresponding components of two vectors and sums them up.The dot product of two vectors \(\mathbf{a} = \langle a_1, a_2 \rangle \) and \\(\mathbf{b} = \langle b_1, b_2 \rangle \) \ is calculated as follows:

• Multiply each component of the first vector by the corresponding component of the second vector: \(a_1 \times b_1\) and \(a_2 \times b_2\).
• Add these products together: \(a_1b_1 + a_2b_2\).

For example, if you have \(\mathbf{u} = \langle -2, 1 \rangle\) and \(\mathbf{v} = \langle 9, 12 \rangle\), then \\(\mathbf{u} \cdot \mathbf{v} = (-2) \times 9 + 1 \times 12 = -18 + 12 = -6\).The result, \(-6\), is a scalar, not a vector, which means it represents a single quantity derived from the two input vectors. The dot product can offer insight into the angle between vectors and is zero when vectors are perpendicular.

Scalar Multiplication

Scalar multiplication involves the multiplication of a vector by a scalar (a real number). When a vector is multiplied by a scalar, each component of the vector is multiplied by this number.For a vector \(\mathbf{v} = \langle v_1, v_2 \rangle\) and a scalar \(c\), the scalar multiplication \(c\mathbf{v}\) is:

• Multiply each component of the vector by the scalar: \(c \times v_1\) and \(c \times v_2\).

As an example, if \(\mathbf{v} = \langle 3, 4 \rangle\) and the scalar is 3, performing scalar multiplication gives:

• \(3 \times 3 = 9\)
• \(3 \times 4 = 12\)

Thus, the new vector after scalar multiplication is \(3 \mathbf{v} = \langle 9, 12 \rangle\).This operation is used to scale vectors, effectively changing their magnitude without altering their direction unless the scalar is negative, which also reverses the direction.

Vector Components

Vector components are the individual parts or elements of a vector, often expressed in a coordinate system like the Cartesian coordinate plane. A vector in 2D can be written as \(\mathbf{v} = \langle v_1, v_2 \rangle\), indicating it has two components.

Understanding Vector Components

In this notation:

• \(v_1\) is the horizontal (x-axis) component.
• \(v_2\) is the vertical (y-axis) component.

Each of these components describes how much of the vector lies along the respective axis. For instance, given vectors like \(\mathbf{u} = \langle -2, 1 \rangle\) and \(\mathbf{v} = \langle 3, 4 \rangle\), the numbers \(-2\), \(1\), \(3\), and \(4\) are vector components.

Importance of Vector Components

• They help in performing vector addition and subtraction.
• They are crucial for computing dot products and cross products.
• They allow visualization and graphical representation of vectors in a space.

Understanding how these components relate to each other is essential for mastering vector operations, as every operation builds upon these fundamental units.