Problem 54 - Precalculus with Limits

Question

In Exercises 53-58, determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, parallel, or neither. \(\mathbf{u} = \langle 3, 15 \rangle\) \(\mathbf{v} = \langle -1, 5 \rangle\)

Step-by-Step Solution

Answer

The vectors \(\mathbf{u}\) and \(\mathbf{v}\) are neither orthogonal nor parallel.

Step 1: Calculate the dot product of \(\mathbf{u}\) and \(\mathbf{v}\)

The dot product of two vectors \(\mathbf{a} = \langle a_1, a_2 \rangle\) and \(\mathbf{b} = \langle b_1, b_2 \rangle\) is given by \(\mathbf{a} \cdot \mathbf{b} = a_1*b_1 + a_2*b_2\). Now, calculate the dot product of \(\mathbf{u}\) and \(\mathbf{v}\): \(\mathbf{u} \cdot \mathbf{v} = (3)*(-1) + (15)*(5) = -3 + 75 = 72\).

Step 2: Check if they are orthogonal

Two vectors are orthogonal if their dot product equals zero. Since the dot product of \(\mathbf{u}\) and \(\mathbf{v}\) equals 72 which is not equal to zero, therefore \(\mathbf{u}\) and \(\mathbf{v}\) are not orthogonal.

Step 3: Check if they are parallel

Two vectors are parallel if they are scalar multiples of each other. From the given vectors, you can see that \(\mathbf{v}\) is not a scalar multiple of \(\mathbf{u}\), or vice versa. So, \(\mathbf{u}\) and \(\mathbf{v}\) are not parallel.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector mathematics that provides valuable information about the relationship between two vectors. It is calculated by taking the product of corresponding components of two vectors and then summing them up. Mathematically, for two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), the dot product is given by:\[ \mathbf{a} \cdot \mathbf{b} = a_1 \cdot b_1 + a_2 \cdot b_2 \]

• This operation yields a scalar (a single number), unlike vector addition or subtraction which results in another vector.
• The dot product not only tells us about the geometric relationship between vectors but can also help in determining angles.
• If the dot product is zero, it indicates orthogonality, meaning the vectors are perpendicular.

In the given example, the calculation of \( \mathbf{u} \cdot \mathbf{v} \) results in 72, which provides us clues about the connection between these vectors. Since the result is not zero, they are not orthogonal. Let's explore this more in the next section.

Orthogonal Vectors

Orthogonal vectors are vectors that meet at right angles to each other. If two vectors are orthogonal, their dot product will be zero. This relationship is extremely important for simplifying problems involving vectors, and orthogonality can be used in fields like computer graphics and physics. Here’s why:

• When vectors are orthogonal, they are completely independent in terms of direction.
• Orthogonal vectors form the basis for defining coordinate systems, like Cartesian coordinates.
• They strike a balance where neither influences the other along its direction, providing clarity in analysis.

In our scenario with \( \mathbf{u} = \langle 3, 15 \rangle \) and \( \mathbf{v} = \langle -1, 5 \rangle \), their dot product is 72, thus confirming they are not orthogonal. This means that the vectors do not meet at a right angle. Understanding orthogonality helps in exploring vector spaces and optimization problems.

Parallel Vectors

Parallel vectors point in the same direction or exactly opposite directions and can be expressed as scalar multiples of each other. For vectors to be parallel:

• One vector should be a scalar multiple of the other. Mathematically, \( \mathbf{a} = k \mathbf{b} \) where \( k \) is a scalar.
• Parallel vectors are collinear which means they lie on the same line, extending forever in both directions.
• The geometric visualization is straightforward; two arrows either on top of each other or in reverse directions.

For this exercise, the vectors \( \mathbf{u} = \langle 3, 15 \rangle \) and \( \mathbf{v} = \langle -1, 5 \rangle \) do not satisfy this condition. There's no scalar \( k \) such that \( \mathbf{u} = k \mathbf{v} \), thus they are not parallel. This understanding ensures the vectors occupy different lines, adding diversity to their spatial representation and utility in various calculations.