Math Problem Statement
Find a unit vector that is orthogonal to both u and v.
u
=
1, −3, 5
v
=
4, 2, 3
Solution
To find a unit vector that is orthogonal to both and , we need to first calculate the cross product , which will give us a vector orthogonal to both and . Then, we will normalize this vector to obtain a unit vector.
Step 1: Find the cross product
Given:
The cross product is:
Expanding the determinant:
So, the vector orthogonal to both and is:
Step 2: Normalize the vector to obtain a unit vector
To normalize , we need to divide it by its magnitude , which is given by:
The unit vector is:
Or, simplifying the components:
This is the unit vector orthogonal to both and .
Would you like any further details or have any questions?
Here are 5 related questions to consider:
- How do you find the magnitude of a vector in three dimensions?
- What does it mean for two vectors to be orthogonal?
- How can you verify that a vector is orthogonal to two given vectors?
- What is the relationship between the cross product and vector areas?
- How do you find the direction of the cross product vector?
Tip: A vector is orthogonal to another if their dot product is zero. This property can be useful to verify your results.
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Math Problem Analysis
Mathematical Concepts
Vectors
Cross Product
Unit Vectors
Formulas
Cross product formula
Unit vector formula
Theorems
-
Suitable Grade Level
High School
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