Math Problem Statement

Given the vectors u = <1, 2, 3> and v = <2, 1, 1>. Express u as the sum u = p + n, where p is parallel to v and n is orthogonal to v.

Solution

To express the vector u\mathbf{u} as the sum u=p+n\mathbf{u} = \mathbf{p} + \mathbf{n}, where p\mathbf{p} is parallel to v\mathbf{v} and n\mathbf{n} is orthogonal to v\mathbf{v}, we can follow these steps:

  1. Calculate the projection of u\mathbf{u} onto v\mathbf{v} to find p\mathbf{p}: projvu=uvvvv\text{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}

    • Calculate the dot products: uv=12+21+31=2+2+3=7\mathbf{u} \cdot \mathbf{v} = 1 \cdot 2 + 2 \cdot 1 + 3 \cdot 1 = 2 + 2 + 3 = 7 vv=22+11+11=4+1+1=6\mathbf{v} \cdot \mathbf{v} = 2 \cdot 2 + 1 \cdot 1 + 1 \cdot 1 = 4 + 1 + 1 = 6

    • Now calculate the projection: projvu=76v=762,1,1=146,76,76=73,76,76\text{proj}_{\mathbf{v}} \mathbf{u} = \frac{7}{6} \mathbf{v} = \frac{7}{6} \langle 2, 1, 1 \rangle = \left\langle \frac{14}{6}, \frac{7}{6}, \frac{7}{6} \right\rangle = \left\langle \frac{7}{3}, \frac{7}{6}, \frac{7}{6} \right\rangle

    So, p=73,76,76\mathbf{p} = \left\langle \frac{7}{3}, \frac{7}{6}, \frac{7}{6} \right\rangle.

  2. Calculate n\mathbf{n}: n=up=1,2,373,76,76\mathbf{n} = \mathbf{u} - \mathbf{p} = \langle 1, 2, 3 \rangle - \left\langle \frac{7}{3}, \frac{7}{6}, \frac{7}{6} \right\rangle

    • Perform the subtraction: n=173,276,376\mathbf{n} = \left\langle 1 - \frac{7}{3}, 2 - \frac{7}{6}, 3 - \frac{7}{6} \right\rangle =3373,12676,18676=43,56,116= \left\langle \frac{3}{3} - \frac{7}{3}, \frac{12}{6} - \frac{7}{6}, \frac{18}{6} - \frac{7}{6} \right\rangle = \left\langle -\frac{4}{3}, \frac{5}{6}, \frac{11}{6} \right\rangle

Thus, the decomposition of u\mathbf{u} is: u=73,76,76+43,56,116\mathbf{u} = \left\langle \frac{7}{3}, \frac{7}{6}, \frac{7}{6} \right\rangle + \left\langle -\frac{4}{3}, \frac{5}{6}, \frac{11}{6} \right\rangle

Summary:

  • p=73,76,76\mathbf{p} = \left\langle \frac{7}{3}, \frac{7}{6}, \frac{7}{6} \right\rangle (parallel to v\mathbf{v})
  • n=43,56,116\mathbf{n} = \left\langle -\frac{4}{3}, \frac{5}{6}, \frac{11}{6} \right\rangle (orthogonal to v\mathbf{v})

Would you like further details or have any questions? Here are some follow-up questions to consider:

  1. How do you calculate the angle between two vectors?
  2. What are the geometric interpretations of parallel and orthogonal vectors?
  3. Can you derive the formula for the projection of one vector onto another?
  4. What properties do orthogonal vectors have in a vector space?
  5. How would you generalize this method for higher dimensions?

Tip: Always check that the dot product of n\mathbf{n} and v\mathbf{v} is zero to confirm orthogonality.

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Math Problem Analysis

Mathematical Concepts

Vector Decomposition
Linear Algebra

Formulas

Projection of u onto v: proj_v(u) = (u · v / v · v) * v
Orthogonal component: n = u - p

Theorems

Projection theorem

Suitable Grade Level

Grades 10-12