https://file.aiquickdraw.com/mathbot/file/1726781070270mnzc83ev.jpg

The image presents a problem asking to find the values of \( x \) such that vector \( \mathbf{a} \) is orthogonal (perpendicular) to vector \( \mathbf{b} \). The vectors are defined as:

\[
\mathbf{a} = \begin{pmatrix} x \\ 2 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} x - 3 \\ 1 \end{pmatrix}
\]

For two vectors to be orthogonal, their dot product must equal zero. The dot product of vectors \( \mathbf{a} \) and \( \mathbf{b} \) is:

\[
\mathbf{a} \cdot \mathbf{b} = (x)(x - 3) + (2)(1) = 0
\]

Let's solve the equation:

\[
x(x - 3) + 2 = 0
\]
\[
x^2 - 3x + 2 = 0
\]

Now, factorize the quadratic equation:

\[
(x - 1)(x - 2) = 0
\]

The solutions are:

\[
x = 1 \quad \text{or} \quad x = 2
\]

Now, substitute the solutions back into the answer choices:

- For \( x = 1 \), vector \( \mathbf{a} = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \) and vector \( \mathbf{b} = \begin{pmatrix} -2 \\ 1 \end{pmatrix} \).
- For \( x = 2 \), vector \( \mathbf{a} = \begin{pmatrix} 2 \\ 2 \end{pmatrix} \) and vector \( \mathbf{b} = \begin{pmatrix} -1 \\ 1 \end{pmatrix} \).

It seems that **option D (x = 1, x = 2)** is correct based on the problem setup.

Would you like further details on the calculations or to clarify anything?

Here are five related questions:
1. How do you calculate the dot product of two vectors?
2. What is the geometric interpretation of orthogonal vectors?
3. How do you factor quadratic equations?
4. How can you check if two vectors are perpendicular without calculating the dot product?
5. What are some applications of orthogonal vectors in real-life problems?

**Tip:** Orthogonal vectors are useful in computer graphics to represent directions that are independent of each other, such as the x and y axes in a 2D space.

Reiknaðu x þannig að vigurinn a sé hornrétt á vigurinn b.

Vectors: 
a = (x, 2)
b = (x-3, 1)
Answer choices:
a) x = -1, x = 2
b) x = 1, x = -2
c) x = -2, x = 3
d) x = 1, x = 2 (Correct)
e) x = -3, x = 2

Learn how to solve for x so that vector a = (x, 2) is orthogonal to vector b = (x-3, 1). This problem involves understanding the dot product of two vectors and solving quadratic equations. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation