Problem 74 - Precalculus with Calculus Previews

Question

In Problems \(65-76, \mathbf{u}=\langle 1,-3,2\rangle, \mathbf{v}=\langle-1,1,1\rangle\), and \(\mathbf{w}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. $$ (\mathbf{v} \cdot \mathbf{w}) \mathbf{u} $$

Step-by-Step Solution

Answer

The resulting vector is \( \langle 13, -39, 26 \rangle \).

Step 1: Find the Dot Product

To solve this problem, we first need to calculate the dot product \( \mathbf{v} \cdot \mathbf{w} \). We use the formula for the dot product: \( \mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 + v_3 w_3 \). For vectors \( \mathbf{v} = \langle -1, 1, 1 \rangle \) and \( \mathbf{w} = \langle 2, 6, 9 \rangle \), we perform:\[(-1) \cdot 2 + 1 \cdot 6 + 1 \cdot 9 = -2 + 6 + 9 = 13\]Thus, \( \mathbf{v} \cdot \mathbf{w} = 13 \).

Step 2: Multiply the Scalar by the Vector

Now that we have the scalar \( \mathbf{v} \cdot \mathbf{w} = 13 \), we need to multiply it with the vector \( \mathbf{u} = \langle 1, -3, 2 \rangle \). This involves multiplying each component of \( \mathbf{u} \) by 13:\[13 \cdot \langle 1, -3, 2 \rangle = \langle 13 \cdot 1, 13 \cdot (-3), 13 \cdot 2 \rangle = \langle 13, -39, 26 \rangle\]Therefore, the resulting vector is \( \langle 13, -39, 26 \rangle \).

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector algebra. It involves taking two vectors and returning a single scalar value.
The formula for calculating the dot product of two vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \) is as follows:

• \( \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 \)

This operation is useful for determining the angle between two vectors or checking if they are orthogonal (perpendicular). If the dot product is zero, the vectors are perpendicular. In our example, we calculated the dot product \( \mathbf{v} \cdot \mathbf{w} = 13 \), which is a scalar.

Scalar Multiplication

Scalar multiplication is another crucial concept in vector algebra. It involves multiplying each component of a vector by a scalar (a single number). This operation changes the magnitude of the vector but not its direction, unless the scalar is negative, which reverses the direction.
For a vector \( \mathbf{u} = \langle u_1, u_2, u_3 \rangle \) and a scalar \( k \), the scalar multiplication is:

• \( k \cdot \mathbf{u} = \langle k \cdot u_1, k \cdot u_2, k \cdot u_3 \rangle \)

In our problem, we multiplied the scalar \( 13 \) by the vector \( \mathbf{u} = \langle 1, -3, 2 \rangle \), resulting in \( \langle 13, -39, 26 \rangle \). This step shows how scalars can stretch or shrink vectors.

Vectors

Vectors are mathematical objects characterized by both a magnitude and a direction. They are essential in many areas of mathematics and science, especially in physics for representing quantities like force and velocity.
A vector with three components can be written as \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \). Each component represents a part of the vector along one of the three Cartesian axes \( (x, y, z) \).

• The magnitude or length of the vector can be calculated using the formula: \( ||\mathbf{v}|| = \sqrt{v_1^2 + v_2^2 + v_3^2} \).
• Vectors are often added together or multiplied using operations such as the dot product or scalar multiplication.

Understanding vectors and their operations is crucial for solving various applied and theoretical problems.

Precalculus

Precalculus lays the foundation for calculus and includes the study of functions, sequences, and series. It also covers trigonometry and algebraic structures like vectors.
In precalculus, students explore concepts that prepare them for calculus-level problems, which involve limits, derivatives, and integrals. The study includes vector operations such as dot products and scalar multiplication.

• These operations help in understanding physical phenomena and solving practical problems in engineering and physics.
• Precalculus also emphasizes the ability to visualize and manipulate mathematical entities, skills that are critical in higher-level mathematics.

Overall, precalculus is essential for developing the analytical skills needed to tackle complex mathematical challenges.