Math Problem Statement
Find three mutually orthogonal unit vectors in Bold Upper R cubed besides plus or minusi, plus or minusj, and plus or minusk. There are multiple ways to do this and an infinite number of answers. For this problem, we choose a first vector u randomly, choose all but one component of a second vector v randomly, and choose the first component of a third vector w randomly. The other components x, y, and z are chosen so that u, v, and w are mutually orthogonal. Then unit vectors are found based on u, v, and w. Start with uequalsleft angle1,1,2right angle, vequalsleft anglex,minus1,2right angle, and wequalsleft angle1,y,zright angle. Question content area bottom Part 1 The unit vector based on u is left angle nothing comma nothing comma nothing right angle
. (Type exact answers, using radicals as needed.)
Solution
Let's solve this problem step by step.
We are given:
- Vector
- Vector
- Vector
We need to find unit vectors for , , and , meaning we need to normalize them so their magnitudes are 1.
Step 1: Find the unit vector based on
To normalize a vector, we divide it by its magnitude. The magnitude of is:
So, the unit vector based on is:
Thus, the unit vector based on is:
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Vector Normalization
Orthogonality
Formulas
Magnitude of a vector: ||v|| = √(v1^2 + v2^2 + v3^2)
Unit vector: v̂ = v / ||v||
Dot product for orthogonality: u ⋅ v = 0
Theorems
Vector Orthogonality Theorem
Suitable Grade Level
Grades 11-12 and College Level
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