Problem 70 - 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. $$ |\mathbf{w}||2 \mathbf{v}| $$

Step-by-Step Solution

Answer

The product is \(22\sqrt{3}\).

Step 1: Understand the given vectors

We have three vectors: \( \mathbf{u} = \langle 1, -3, 2 \rangle \), \( \mathbf{v} = \langle -1, 1, 1 \rangle \), and \( \mathbf{w} = \langle 2, 6, 9 \rangle \). We need to find the product of the magnitudes of \( \mathbf{w} \) and \( 2\mathbf{v} \).

Step 2: Calculate the magnitude of vector \( \mathbf{w} \)

The formula for the magnitude of a vector \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) is \( |\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \). For \( \mathbf{w} \), this becomes \( |\mathbf{w}| = \sqrt{2^2 + 6^2 + 9^2} = \sqrt{4 + 36 + 81} = \sqrt{121} = 11 \).

Step 3: Calculate vector \( 2\mathbf{v} \)

To find \( 2\mathbf{v} \), multiply each component of \( \mathbf{v} \) by 2: \( 2\mathbf{v} = 2 \times \langle -1, 1, 1 \rangle = \langle -2, 2, 2 \rangle \).

Step 4: Calculate the magnitude of vector \( 2\mathbf{v} \)

Using the same magnitude formula, compute \( |2\mathbf{v}| = \sqrt{(-2)^2 + 2^2 + 2^2} = \sqrt{4 + 4 + 4} = \sqrt{12} = 2\sqrt{3} \).

Step 5: Multiply the magnitudes

Now, calculate the product of the magnitudes: \( |\mathbf{w}||2\mathbf{v}| = 11 \times 2\sqrt{3} = 22\sqrt{3} \).

Key Concepts

Vector Operations

Vectors are mathematical entities that have both a direction and a magnitude. In the realm of vector operations, we focus on actions we can perform using vectors. These include addition, subtraction, and scalar multiplication. Each of these operations follows specific rules and has different results.

• Addition: Simply add corresponding components of two vectors. For example, if we have vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \), their sum is \( \langle a_1 + b_1, a_2 + b_2, a_3 + b_3 \rangle \).
• Subtraction: Subtract corresponding components. Thus, the subtraction of vectors \( \mathbf{a} \) and \( \mathbf{b} \) gives \( \langle a_1 - b_1, a_2 - b_2, a_3 - b_3 \rangle \).
• Scalar Multiplication: This involves multiplying a vector by a scalar value, affecting the magnitude but not the direction, unless the factor is negative.

The operations are essential building blocks for understanding more complex vector concepts.

Scalar Multiplication

Scalar multiplication is a vital concept in vector mathematics and involves multiplying a vector by a scalar, which is simply a real number. This process changes the magnitude of the vector while maintaining its direction unless the scalar is negative.
When multiplying a vector by a scalar, each component of the vector is multiplied by the scalar. For instance, given a vector \( \mathbf{v} = \langle x, y, z \rangle \) and a scalar \( c \), the product \( c \mathbf{v} \) would be the vector \( \langle cx, cy, cz \rangle \).
Applying scalar multiplication can help in scaling vectors to fit different applications, such as stretching or compressing them to desired lengths. It's also a fundamental operation that supports other aspects of vector algebra, such as finding magnitudes.

Dot Product

The dot product, also known as the scalar product, is an operation that takes two vectors and returns a single scalar. This operation reflects the degree to which two vectors are pointing in the same direction.
The formula for the dot product of vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \) is:
\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]
The result is a scalar that is determined by both the magnitudes of the vectors as well as the cosine of the angle between them. Hence, it is very useful for understanding the alignment of vectors:

• The dot product is zero if the vectors are orthogonal (i.e., at a right angle).
• It is positive if the vectors point in a similar direction.
• It is negative when the vectors point in opposite directions.

Knowing how to use the dot product can be helpful for many applications, such as physics to calculate projections or in computer graphics to determine shading.

Vector Properties

Vectors come with various properties that help in understanding how they behave in mathematical operations and real-world applications. Here are some key properties:

• Magnitude: Also known as the length or norm of a vector, it is always a non-negative value. The magnitude of a vector \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) is calculated as \( \sqrt{a_1^2 + a_2^2 + a_3^2} \).
• Direction: Unlike scalars which only have magnitude, vectors also define a direction. This is usually expressed as unit vectors that indicate the direction of the vector.
• Equality: Two vectors are equal if their corresponding components are equal.
• Zero Vector: A vector with zero magnitude (e.g. \( \langle 0, 0, 0 \rangle \)) has no direction.

Understanding vector properties allows better comprehension of their behavior under various operations and is fundamental in both theoretical and applied fields like engineering, physics, and computer science.