Problem 27 - Precalculus

Question

Determine whether each pair of vectors is orthogonal. $$\langle 6,-4\rangle \text { and }\langle-6,-9\rangle$$

Step-by-Step Solution

Answer

Yes, the vectors are orthogonal.

Step 1: Understand Orthogonality

Two vectors are orthogonal if their dot product is zero. The dot product of two vectors \( \langle a_1, a_2 \rangle \) and \( \langle b_1, b_2 \rangle \) is given by \( a_1 \cdot b_1 + a_2 \cdot b_2 \).

Step 2: Calculate the Dot Product

For vectors \( \langle 6,-4 \rangle \) and \( \langle -6,-9 \rangle \), the dot product is calculated as follows:\[ 6 \cdot (-6) + (-4) \cdot (-9) \] which simplifies to:\[ -36 + 36 = 0 \]

Step 3: Determine Orthogonality

Since the dot product calculated is 0, the vectors \( \langle 6,-4 \rangle \) and \( \langle -6,-9 \rangle \) are orthogonal.

Key Concepts

Dot Product

The dot product is a fundamental concept used in vector mathematics to find how "aligned" two vectors are. When you take the dot product of two vectors, you multiply corresponding components and sum the results. For example, with vectors

• \( \langle a_1, a_2 \rangle \)
• \( \langle b_1, b_2 \rangle \),

the formula for the dot product is \( a_1 \cdot b_1 + a_2 \cdot b_2 \).

When calculating the dot product, the result is a single number. This number describes the magnitude and directional relationship between the two vectors. A helpful way to visualize this is by imagining how much one vector contributes directionally to another.

One special case of the dot product is when it equals zero. This is significant and indicates that the vectors are orthogonal (or perpendicular to each other)."

Orthogonal Vectors

Orthogonal vectors are vectors that meet at a right angle, meaning they are perpendicular. The condition for two vectors to be orthogonal is that their dot product is zero.

This makes physical and geometrical sense. If two vectors are orthogonal, neither points in the direction of the other. This is why in the

• dot product calculation

any positive or negative contributions from vector components end up cancelling each other out, leading to zero.

In our example, the vectors \( \langle 6, -4 \rangle \) and \( \langle -6, -9 \rangle \) have their dot product calculated as zero:

• \(6 \cdot (-6) = -36\)
• \((-4) \cdot (-9) = 36\)

So, adding these results yields 0, confirming the vectors are orthogonal.

Precalculus Vectors

In precalculus, vectors are introduced to lay the groundwork for various applications in calculus and physics. Vectors are mathematical objects that have both magnitude and direction. They are usually represented in two-dimensional form, like \( \langle x, y \rangle \).

Understanding vectors is crucial because they easily model physical quantities, such as force, velocity, and others which can be represented by both a direction and a magnitude. Precalculus focuses on teaching how to manipulate these vectors through operations such as addition, subtraction, and finding dot products.

These foundational concepts allow students to understand more advanced topics in mathematics and science. By mastering vector operations, including the calculation of dot products to find orthogonal vectors, students set a strong base for delving deeper into the world of mathematics.