Problem 55 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the value of \(k\) such that the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal. $$\begin{aligned} &\mathbf{u}=-3 k \mathbf{i}+2 \mathbf{j}\\\ &\mathbf{v}=-6 \mathbf{i} \end{aligned}$$

Step-by-Step Solution

Answer

The value of \(k\) that makes the vectors \(\mathbf{u}\) and \(\mathbf{v}\) orthogonal is 0.

Step 1: Set up the vectors

First, we write down the vectors \(\mathbf{u}\) and \(\mathbf{v}\). We have \(\mathbf{u}=-3k\mathbf{i}+2\mathbf{j}\) and \(\mathbf{v}=-6\mathbf{i}\). Note that vector \(\mathbf{v}\) has no j component.

Step 2: Calculate the dot product

Next, we calculate the dot product of \(\mathbf{u}\) and \(\mathbf{v}\). The dot product of two vectors is calculated as the sum of the product of their corresponding components. Hence, \(\mathbf{u} \cdot \mathbf{v} = (-3k) (-6) + 2*0 = 18k\). The second term is zero because \(\mathbf{v}\) has no j component.

Step 3: Set the dot product equal to zero

Since \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, their dot product must be zero. Therefore, we set 18k equal to zero. This gives us the equation 18k = 0.

Step 4: Solve for k

Finally, we solve the equation 18k = 0 for k. Dividing both sides by 18 gives us k = 0.

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental way to multiply two vectors. This operation combines two vectors to produce a single scalar quantity. To compute the dot product, you multiply each pair of corresponding components of the vectors and then sum these products. For example, if you have two vectors

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

Then the dot product \( \mathbf{a} \cdot \mathbf{b} \) is:

• \( a_1 \, b_1 + a_2 \, b_2 \)

If one of the vectors is missing a component, as in our example where \( \mathbf{v} = -6 \mathbf{i} \), then that component is treated as zero, simplifying the dot product calculation. Hence, only the relevant non-zero components are multiplied together.

Vector Components

Vectors are mathematical objects that have both magnitude and direction. They can be represented in terms of their components along different axes. For instance, in two dimensions, vectors are typically expressed in terms of the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \), which represent the x and y axes, respectively.

• \( \mathbf{u} = -3k \mathbf{i} + 2 \mathbf{j} \)
• \( \mathbf{v} = -6 \mathbf{i} \)

In the above vectors, the coefficients of \( \mathbf{i} \) and \( \mathbf{j} \) are called components. These components determine the direction and length of the vector in each particular axis.
In our example, vector \( \mathbf{u} \) has both an \( \mathbf{i} \) component and a \( \mathbf{j} \) component, whereas \( \mathbf{v} \) only has an \( \mathbf{i} \) component. Understanding vector components is crucial for tasks such as calculating the dot product or determining if two vectors are orthogonal.

Solving Equations

Equations are mathematical statements that assert the equality of two expressions. Solving an equation involves finding the values of unknown variables that make the equation true. In algebra, this often involves isolating the variable on one side of the equation.
In the step-by-step solution to our exercise, we had the equation

• \( 18k = 0 \)

This equation results from setting the dot product of two orthogonal vectors equal to zero. To solve \( 18k = 0 \), you isolate \( k \) by dividing both sides by 18:

• \( k = \frac{0}{18} = 0 \)

The solution \( k = 0 \) indicates the specific condition under which the vectors \( \mathbf{u} \) and \( \mathbf{v} \) become orthogonal. This reflects the importance of solving equations in vector analysis and in finding particular geometric conditions like orthogonality.