Problem 1 - Algebra and Trigonometry

Question

Let \(\mathbf{a}=\left\langle a_{1}, a_{2}\right\rangle\) and \(\mathbf{b}=\left\langle b_{1}, b_{2}\right\rangle\) be nonzero vectors in the plane, and let \(\theta\) be the angle between them. The dot product of a and b is defined by \(\mathbf{a} \cdot \mathbf{b}=\) ______ The dot product of two vectors is a ______, not a vector.

Step-by-Step Solution

Answer

The dot product is \(a_1b_1 + a_2b_2\). It is a scalar, not a vector.

Step 1: Define the Dot Product Formula

The dot product of two vectors \(\mathbf{a} = \langle a_1, a_2 \rangle\) and \(\mathbf{b} = \langle b_1, b_2 \rangle\) in the plane is given by the formula \(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2\). This formula involves multiplying the corresponding components of the two vectors and then adding those products together.

Step 2: Dot Product as a Real Number

The result of a dot product \(\mathbf{a} \cdot \mathbf{b}\) is a scalar, which is a real number. It is not a vector, as the operation combines the components of the vectors into a single value.

Key Concepts

Vectors

Vectors are mathematical objects that have both magnitude and direction. They are used in various fields such as physics, engineering, and computer science to describe quantities that are not only about how much, but also about which way.
For example, a vector can represent velocity, force, or displacement. In a two-dimensional plane, a vector is often written as \( \mathbf{a} = \langle a_1, a_2 \rangle \), where \( a_1 \) and \( a_2 \) are the components of the vector along the x and y axes, respectively.
These components can be visualized as the legs of a right triangle, where the vector is the hypotenuse. This makes it easier to work with vectors graphically by breaking them into their horizontal and vertical parts.

Angle between Vectors

The angle between two vectors can provide insights into their directional relationship. Understanding this angle helps determine how two vectors are oriented with respect to each other in space.
To find the angle \( \theta \) between two vectors \( \mathbf{a} \) and \( \mathbf{b} \), you can use their dot product along with their magnitudes. The formula is:

• \( \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|} \)

Here, \( \|\mathbf{a}\| \) and \( \|\mathbf{b}\| \) are the magnitudes of the vectors, which are found using the Pythagorean theorem: \( \|\mathbf{a}\| = \sqrt{a_1^2 + a_2^2} \) and \( \|\mathbf{b}\| = \sqrt{b_1^2 + b_2^2} \).
The angle \( \theta \) helps determine if the vectors are:

• Perpendicular, when \( \theta = 90^\circ \)
• Opposite in direction, when \( \theta = 180^\circ \)
• In the same direction, when \( \theta = 0^\circ \)

Scalar Product

The scalar product, also known as the dot product, is a way to multiply two vectors to result in a single scalar value. This operation highlights the contribution of one vector in the direction of another.
The formula for the dot product of vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \) is:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \)

By multiplying corresponding components of each vector and summing the results, the dot product gives insight into how parallel the vectors are.
A key property of the dot product is its relationship with the cosine of the angle between the vectors, expressed as:\( \mathbf{a} \cdot \mathbf{b} = \| \mathbf{a} \| \| \mathbf{b} \| \cos \theta \). This equation allows us to derive the angle between the vectors if necessary.

Real Number Result

When you perform the dot product operation on two vectors, the result is a scalar, a real number. Unlike vector multiplication, which can yield a vector, the dot product reduces the information from both vectors into a single value.
This real number has significant importance:

• If the result is positive, it indicates that the vectors are pointing in generally the same direction.
• If the result is negative, it suggests that the vectors are pointing in opposite directions.
• If the dot product is zero, the vectors are perpendicular to each other.

In visual or graphical problems, interpreting the real number result from a dot product can help determine angles, optimize projections, and even solve real-world problems involving directions and magnitudes.