Problem 27 - Contemporary Precalculus

Question

Find a real number \(k\) such that the two vectors are orthogonal. $$\mathbf{i}-\mathbf{j}, k \mathbf{i}+\sqrt{2} \mathbf{j}$$

Step-by-Step Solution

Answer

Answer: \(\sqrt{2}\)

Step 1: Calculate the dot product of the given vectors

First, let's calculate the dot product of the given vectors: \((\mathbf{i}-\mathbf{j}) \cdot (k \mathbf{i}+\sqrt{2} \mathbf{j})\). The dot product is the sum of the product of their corresponding components: \((1 \cdot k) + ((-1) \cdot \sqrt{2})\).

Step 2: Set the dot product equal to zero

Since the dot product of orthogonal vectors is equal to 0, we can set the dot product we calculated in Step 1 equal to 0: \((1 \cdot k) + ((-1) \cdot \sqrt{2}) = 0\).

Step 3: Solve for \(k\)

Now we need to isolate \(k\) by adding \(\sqrt{2}\) to both sides of the equation: \(k = \sqrt{2}\). Therefore, the real number \(k\) that makes the given vectors orthogonal is \(\boxed{\sqrt{2}}\).

Key Concepts

Dot Product

The dot product is a fundamental operation when working with vectors. It is also known as the scalar product, and it involves two vectors resulting in a single scalar quantity. This operation combines the corresponding components of the vectors.
To calculate the dot product of two vectors, each element of the vectors is multiplied with its counterpart in the other vector and then all the products are summed up.For example, if we have two vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \), their dot product is computed as:

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

In this exercise, we used the vectors \( \mathbf{i} - \mathbf{j} \) and \( k \mathbf{i} + \sqrt{2} \mathbf{j} \) and calculated their dot product as \((1 \cdot k) + ((-1) \cdot \sqrt{2})\). This step is crucial, as determining whether vectors are orthogonal depends on the numerical outcome of this calculation.

Vectors

Vectors are mathematical objects that represent both a direction and a magnitude. They are widely used in geometry, physics, and engineering to describe quantities that have both direction and strength, like force or velocity.
Vectors can be easily visualized as arrows; the length of the arrow indicates the magnitude while the direction of the arrow shows the direction.A vector \( \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} \) consists of two components, \( v_1 \) and \( v_2 \), which correspond to the coefficient of the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) respectively in a 2-dimensional space.
Orthogonal vectors are pairs whose directions are at right angles to each other. When vectors are orthogonal, their dot product is zero. This is crucial for determining the orthogonality of vectors simply by calculation.Working with vectors requires you to understand how to manipulate and combine these components carefully.

Solving Equations

Solving equations is the process of finding which values satisfy a given mathematical statement. It often involves isolating a variable or finding the unknowns that make the statement true.
In many problems, especially those involving vectors and physics, solving equations helps find specific conditions or variables.In this exercise, we found real number \( k \) such that the vectors are orthogonal by solving the equation derived from their dot product:

• \((1 \cdot k) + ((-1) \cdot \sqrt{2}) = 0\)

To solve, we isolate \( k \) by adding \( \sqrt{2} \) to both sides:

• \( k = \sqrt{2} \)

This straightforward algebraic manipulation shows a key technique in solving equations: balancing operations to uncover the unknown variable, crucial for determining the specific relationships described in mathematical problems.