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 angle​x,minus​1,2right angle​, and wequalsleft angle​1,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 u=1,1,2u = \langle 1, 1, 2 \rangle
  • Vector v=x,1,2v = \langle x, -1, 2 \rangle
  • Vector w=1,y,zw = \langle 1, y, z \rangle

We need to find unit vectors for uu, vv, and ww, meaning we need to normalize them so their magnitudes are 1.

Step 1: Find the unit vector based on u=1,1,2u = \langle 1, 1, 2 \rangle

To normalize a vector, we divide it by its magnitude. The magnitude of uu is:

u=12+12+22=1+1+4=6\| u \| = \sqrt{1^2 + 1^2 + 2^2} = \sqrt{1 + 1 + 4} = \sqrt{6}

So, the unit vector based on uu is:

u^=uu=161,1,2=16,16,26\hat{u} = \frac{u}{\| u \|} = \frac{1}{\sqrt{6}} \langle 1, 1, 2 \rangle = \langle \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}} \rangle

Thus, the unit vector based on uu is:

u^=16,16,26\hat{u} = \langle \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}} \rangle


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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