Problem 73 - Precalculus with Calculus Previews

Question

Find a scalar \(c\) so that the given vectors are orthogonal. $$ \mathbf{u}=2 \mathbf{i}-c \mathbf{j}, \mathbf{v}=3 \mathbf{i}+2 \mathbf{j} $$

Step-by-Step Solution

Answer

The scalar \( c \) is 3.

Step 1: Understand Orthogonality

Two vectors are orthogonal if their dot product is zero. The dot product of vectors \( \mathbf{u} = 2\mathbf{i} - c\mathbf{j} \) and \( \mathbf{v} = 3\mathbf{i} + 2\mathbf{j} \) needs to be calculated and set to zero to find the scalar \( c \).

Step 2: Compute the Dot Product

The dot product of vectors \( \mathbf{u} \) and \( \mathbf{v} \) is calculated as: \[ \mathbf{u} \cdot \mathbf{v} = (2)(3) + (-c)(2) \] Simplifying gives: \[ \mathbf{u} \cdot \mathbf{v} = 6 - 2c \]

Step 3: Set Dot Product to Zero

To make the vectors orthogonal, set the dot product equal to zero:\[ 6 - 2c = 0 \]

Step 4: Solve for \( c \)

Solve the equation from the previous step:\[ 6 = 2c \]Divide both sides by 2 to isolate \( c \):\[ c = 3 \]

Key Concepts

Dot Product

The dot product is a fundamental operation in vector algebra. It's a way to multiply two vectors to obtain a scalar value. For vectors \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} \), the dot product is given by:\[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2\]This operation helps to determine the orthogonality of vectors. If the dot product equals zero, the vectors are orthogonal, meaning they meet at a right angle. Understanding this concept is crucial for solving problems involving the angle and perpendicularity of vectors.

• The dot product results in a scalar, not a vector.
• If the dot product is zero, vectors are orthogonal.
• It simplifies vector calculations that involve angles.

Scalar Multiplication

Scalar multiplication occurs when a vector is multiplied by a scalar (a regular number). This operation scales the vector without changing its direction unless the scalar is negative, which reverses its direction. Given a vector \( \mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j} \) and a scalar \( c \), the scalar multiplication is given by:\[c \cdot \mathbf{v} = cv_1\mathbf{i} + cv_2\mathbf{j}\]For example, multiplying a vector by 2 would double the length of the vector while maintaining its direction. Scalar multiplication is important in adjusting the magnitude of vectors according to the problem's requirements.

• Does not change the vector's direction unless the scalar is negative.
• Used to adjust the length (magnitude) of a vector.
• Occurs frequently in equations and transformations involving vectors.

Vector Calculations

Vector calculations involve a variety of operations used to analyze vectors' properties and relationships. These calculations are essential for understanding vectors in physics, engineering, and mathematics. Common vector operations include addition, subtraction, and multiplication (both scalar multiplication and the dot product).
Vector addition involves combining two vectors to form a new vector. For two vectors \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} \), their sum is:\[\mathbf{a} + \mathbf{b} = (a_1 + b_1)\mathbf{i} + (a_2 + b_2)\mathbf{j}\]

• Essential in understanding motion and forces in physics.
• Helps solve geometric problems related to direction and length.
• Foundation for more complex vector analysis, including vector spaces and transformations.

These operations allow for a comprehensive analysis of vectors and their applications, making them a cornerstone of many scientific fields.