Problem 10 - Trigonometry

Question

Find each of the following dot products $\langle 6,6\rangle \cdot\langle-3,5\rangle

Step-by-Step Solution

Answer

The dot product is 12.

Step 1: Identify the vectors

The two vectors we need to work with are \( \langle 6, 6 \rangle \) and \( \langle -3, 5 \rangle \). Each vector has two components: the first vector has components 6 and 6, while the second vector has components -3 and 5.

Step 2: Apply the formula for dot product

The formula for the dot product of two vectors \( \langle a_1, a_2 \rangle \) and \( \langle b_1, b_2 \rangle \) is given by \( a_1 \cdot b_1 + a_2 \cdot b_2 \). In this case: \[ \langle 6, 6 \rangle \cdot \langle -3, 5 \rangle = (6 \times -3) + (6 \times 5) \].

Step 3: Calculate each multiplication

Carry out the multiplications separately. First, calculate \( 6 \times -3 = -18\). Then, calculate \( 6 \times 5 = 30\).

Step 4: Add the products

Now add the two products you calculated: \(-18 + 30\).

Step 5: Obtain the final result

After adding the two products, you get: \( -18 + 30 = 12 \). Thus, the dot product of the vectors \( \langle 6, 6 \rangle \) and \( \langle -3, 5 \rangle \) is 12.

Key Concepts

Vectors

Vectors are fundamental components in mathematics and physics. Think of a vector as a way to describe a direction and magnitude in space. In our exercise, there's a vector \( \langle 6, 6 \rangle \) which indicates that you move 6 units in one direction and 6 in another. Similarly, the vector \( \langle -3, 5 \rangle \) implies movement of -3 units and 5 units in their respective directions. These directions can be visualized graphically on a two-dimensional plane. Vectors have multiple components corresponding to the dimensions of the space they are in. In this problem, the vectors have two components because they exist in a 2-dimensional space. This allows them to be easily represented in a plane. Understanding vectors as pairs of movements makes them more intuitive and aids in visualizing problems involving more complex multi-dimensional spaces.

Multiplication

Multiplication plays a pivotal role in finding the dot product of two vectors. It's important to grasp that we are not multiplying the vectors as whole objects, but rather their respective components. Each component in the first vector is multiplied by the corresponding component in the second vector. In our example, we perform two separate multiplications:

• First, the initial components: \( 6 \times -3 = -18 \)
• Then, the second components: \( 6 \times 5 = 30 \)

Obtaining these products is critical because they represent how much each vector contributes to their shared magnitude in the space. You can think of it as measuring alignment along each dimension.

Addition

Once you have the products of the corresponding components from the vectors, addition is used to find the final dot product. This step involves summing the individual products calculated from the component multiplications.In this instance, we're adding:

• \( -18 \)
• \( 30 \)

Putting these together results in \( -18 + 30 = 12 \). This sum is the dot product of the vectors. Adding these products gives a scalar or single value which signifies how much the vectors align in terms of magnitude and direction in the dimensional space. A positive dot product, like ours, indicates some degree of alignment between the vectors.