Problem 52 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the value of \(k\) such that the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal. $$\begin{aligned} &\mathbf{u}=8 \mathbf{i}+4 \mathbf{j}\\\ &\mathbf{v}=2 \mathbf{i}-k \mathbf{j} \end{aligned}$$

Step-by-Step Solution

Answer

The value of \(k\) that makes vectors \(\mathbf{u}\) and \(\mathbf{v}\) orthogonal is \(k = 4\).

Step 1: Write out vector \(\mathbf{u}\) and vector \(\mathbf{v}\)

\(\mathbf{u}\) = \(8\mathbf{i} + 4\mathbf{j}\) and \(\mathbf{v}\) = \(2\mathbf{i} - k\mathbf{j}\)

Step 2: Formulate the dot product

The dot product of \(\mathbf{u}\) and \(\mathbf{v}\) is \(\mathbf{u} \cdot \mathbf{v} = (8 \cdot 2) + (4 \cdot -k)\)

Step 3: Calculate the dot product

Solve for \(\mathbf{u} \cdot \mathbf{v}\) to get \(16 - 4k\)

Step 4: Set the dot product equal to zero and solve for \(k\)

Set \(\mathbf{u} \cdot \mathbf{v} = 0\): We then get \(16 - 4k = 0\). Solving for \(k\) gives us \(k = 4\)

Key Concepts

Understanding the Dot Product

The dot product is a fundamental concept in vector mathematics that helps us determine the relationship between two vectors. It is also known as the scalar product because it results in a scalar (a single number), rather than a vector. To calculate the dot product of two vectors, you multiply corresponding components and then sum the results. For example, if you have two 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 \( \mathbf{a} \cdot \mathbf{b} \) can be calculated using the formula: \[ \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \] In our exercise, we use the dot product to determine if two vectors are orthogonal. Orthogonal vectors have a dot product of zero. Therefore, by setting the dot product equal to zero, we can solve for unknown variables and confirm orthogonality.

Performing Vector Operations

Vectors are essential in representing quantities that have both magnitude and direction, like velocity or force. Vector operations are crucial in physics and engineering, among many other fields. Basic vector operations include addition, subtraction, scalar multiplication, and finding dot products.

• Addition: Add corresponding components of vectors. \( \mathbf{a} + \mathbf{b} = (a_1 + b_1)\mathbf{i} + (a_2 + b_2)\mathbf{j} \)
• Subtraction: Subtract corresponding components. \( \mathbf{a} - \mathbf{b} = (a_1 - b_1)\mathbf{i} + (a_2 - b_2)\mathbf{j} \)
• Scalar Multiplication: Multiply each component by the scalar. \( c\mathbf{a} = (c\times a_1)\mathbf{i} + (c\times a_2)\mathbf{j} \)
• Dot Product: Useful for checking orthogonality as discussed.

By understanding these operations, you'll have a toolkit for tackling various problems involving vectors in both theoretical and applied contexts.

Solving for Variables in Vectors

Finding unknown variables in vector problems often involves setting up equations using vector operations. In our specific problem, we are asked to find the variable \( k \) that makes the vectors orthogonal. Start by computing the dot product of the given vectors: \( \mathbf{u} = 8\mathbf{i} + 4\mathbf{j} \) and \( \mathbf{v} = 2\mathbf{i} - k\mathbf{j} \). The dot product formula gives us: \[ \mathbf{u} \cdot \mathbf{v} = (8 \times 2) + (4 \times -k) = 16 - 4k \] We know that for vectors to be orthogonal, this dot product must equal zero, therefore: \[ 16 - 4k = 0 \] To find \( k \), isolate it on one side of the equation:

• Subtract 16 from both sides: \(-4k = -16\)
• Divide by -4: \(k = 4\)

Solving such equations helps confirm the conditions under which vectors maintain certain relationships, like orthogonality in this case.