Questions: What are orthogonal vectors? Two vectors are orthogonal when the dot product of the vectors is 1 or -1. Two vectors are orthogonal when the angle between the vectors is 90°.

What are orthogonal vectors? Two vectors are orthogonal when the dot product of the vectors is 1 or -1. Two vectors are orthogonal when the angle between the vectors is 90°.
Transcript text: What are orthogonal vectors? Two vectors are orthogonal when the dot product of the vectors is 1 or -1. Two vectors are orthogonal when the angle between the vectors is $90^{\circ}$.
failed

Solution

failed
failed

Solution Steps

Step 1: Definition of Orthogonal Vectors

Orthogonal vectors are defined as two vectors that are perpendicular to each other. This means that the angle between the two vectors is \(90^{\circ}\).

Step 2: Mathematical Condition for Orthogonality

For two vectors \(\mathbf{a}\) and \(\mathbf{b}\) to be orthogonal, their dot product must be zero. Mathematically, this is expressed as: \[ \mathbf{a} \cdot \mathbf{b} = 0 \]

Step 3: Explanation of Dot Product

The dot product of two vectors \(\mathbf{a} = (a_1, a_2, \ldots, a_n)\) and \(\mathbf{b} = (b_1, b_2, \ldots, b_n)\) is calculated as: \[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \cdots + a_nb_n \] If this sum equals zero, the vectors are orthogonal.

Final Answer

Two vectors are orthogonal when the angle between the vectors is \(90^{\circ}\), which is equivalent to their dot product being zero. \(\boxed{\text{Orthogonal vectors have a dot product of zero.}}\)

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
Find the production matrix for the following input-output and demand matrices using the open model. A=[ 0.1 0.3 0.45 0.1 ] D=[ 3 5 ] The production matrix is (Round the final answer to the nearest hundredth as needed. Round all intermediate values to four decimal places as needed.)
failedAnswered
Use hyphens in compound adjectives where necessary. If no hyphens are needed, submit text without making any changes. "Would you like to live in a dormitory on campus, or would you prefer off-campus housing?" the college recruiter asked Manny.
failedAnswered
A baseball player is asked to swing at pitches in sets of four. The player swings at 100 sets of 4 pitches. The probability distribution for making a particular number of hits is given below, Determine the mean for this discrete probability distribution. x 0 1 2 3 4 P(x) 0.02 0.07 0.22 0.27 0.42 A. 4 B. 3 C. 2 D. 3.5
failedAnswered
Write the quadratic equation whose roots are 6 and 1, and whose leading coefficient is 3. (Use the letter x to represent the variable.)
failedAnswered
Solve the equation. 3.4 = -13.6 + (-3.4c) + 1.7c