Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

For what values of \(a, b\), and \(c\) are the three vectors \(\langle a, 0,1\rangle,\langle 0,2, b\rangle\), and \(\langle 1, c, 1\rangle\) mutually orthogonal.

Step-by-Step Solution

Answer

The values are \( a = -1 \), \( b = 0 \), and \( c = 0 \).

Step 1: Set the Dot Product of First and Second Vectors to Zero

For two vectors to be orthogonal, their dot product must be zero. So, start by calculating the dot product of \( \langle a, 0, 1 \rangle \) and \( \langle 0, 2, b \rangle \). This is:\[ a \cdot 0 + 0 \cdot 2 + 1 \cdot b = 0 + 0 + b = b \]Set it to zero: \[ b = 0 \]

Step 2: Set the Dot Product of First and Third Vectors to Zero

Next, calculate the dot product of the vectors \( \langle a, 0, 1 \rangle \) and \( \langle 1, c, 1 \rangle \):\[ a \cdot 1 + 0 \cdot c + 1 \cdot 1 = a + 0 + 1 = a + 1 \]Set it to zero: \[ a + 1 = 0 \]Solve for \( a \):\[ a = -1 \]

Step 3: Set the Dot Product of Second and Third Vectors to Zero

Finally, calculate the dot product of \( \langle 0, 2, b \rangle \) and \( \langle 1, c, 1 \rangle \):\[ 0 \cdot 1 + 2 \cdot c + b \cdot 1 = 0 + 2c + b = 2c + b \]Since \( b = 0 \) from Step 1, substitute it into the equation:\[ 2c + 0 = 0 \]Solve for \( c \):\[ 2c = 0 \]\[ c = 0 \]

Key Concepts

Orthogonal Vectors

Orthogonal vectors are vectors that are "perpendicular" to each other in terms of spatial orientation. When two vectors are orthogonal, their dot product is always zero. This is because orthogonal vectors do not contribute any component to one another along the axes they define. Hence, their interaction in terms of projection is nonexistent, resulting in a dot product of zero. This concept is very useful in linear algebra, particularly when dealing with spaces and transformations.

• If the dot product of two vectors is zero, they are orthogonal.
• It's like forming a right angle when visualizing the vectors in space.

To determine if vectors are mutually orthogonal, every possible pair of vectors must have a zero dot product. This ensures that each vector intersects with every other vector at right angles. This concept is crucial for problems involving orthonormal bases and simplifies vector equations and transformations.

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in linear algebra. It combines two vectors to produce a single scalar value. The dot product of vectors \( \mathbf{u} = \langle u_1, u_2, u_3 \rangle \) and \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \) is calculated as:\[ u_1v_1 + u_2v_2 + u_3v_3 \]Using the dot product, we can find the angle between two vectors and determine orthogonality. A dot product of zero indicates perpendicular vectors.

• Easy to compute with vectors in any dimension.
• Implies orthogonality when the product equals zero.

In solving problems like those involving orthogonal vectors, you repeatedly perform dot product calculations to check for orthogonality. It's a quick way to confirm vectors do not influence each other in any particular direction.

Vector Operations

Vector operations are actions that on vectors, simplify expressions, or solve equations involving vectors. These include addition, subtraction, scaling, and special operations like the dot and cross products. Operations involving vectors are often visually conceivable in terms of combinations or associations of directions and magnitudes.

• Addition: Combine vectors to find resultant magnitude and direction.
• Subtraction: Determine the vector between two endpoints.
• Scaling: Stretch or shrink a vector by multiplying with a scalar.

When aiming for orthogonality, you often manipulate vectors via these operations, playing with their orientation and modifying components. Using vector forms, like \( \langle a, 0, 1 \rangle \), allows you to easily plug values into standard vector equations, making theoretical vector manipulations a straightforward process.