Math Problem Statement

Let u = i + n(j - i) and v = i - j, where n is a constant. Find the angles between u and v.

Solution

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v) Let u=i+n(ji)u = i + n(j - i) and v=ijv = i - j, where nn is a constant. Find the angle(s) between uu and vv.

This question is asking you to find the angle between two vectors uu and vv, given their definitions in terms of other vectors ii and jj.

Solution:

  1. Vectors Given: u=i+n(ji)=(1n)i+nju = i + n(j - i) = (1 - n)i + nj v=ijv = i - j

  2. Dot Product Formula: The angle θ\theta between two vectors uu and vv can be found using the dot product: cos(θ)=uvuv\cos(\theta) = \frac{u \cdot v}{\|u\| \|v\|} Where:

    • uvu \cdot v is the dot product of the two vectors
    • u\|u\| and v\|v\| are the magnitudes (norms) of the vectors.
  3. Dot Product: uv=((1n)i+nj)(ij)u \cdot v = ((1 - n)i + nj) \cdot (i - j) Expanding the dot product: uv=(1n)(ii)+(1n)(ij)+n(ji)+(n)(jj)u \cdot v = (1 - n)(i \cdot i) + (1 - n)(-i \cdot j) + n(j \cdot i) + (-n)(j \cdot j) Simplifying: uv=(1n)(1)+(n)(1)u \cdot v = (1 - n)(1) + (-n)(1) Since ij=0i \cdot j = 0 and ii=jj=1i \cdot i = j \cdot j = 1, we get: uv=1nn=12nu \cdot v = 1 - n - n = 1 - 2n

  4. Magnitudes: Now, calculate the magnitudes u\|u\| and v\|v\|.

    • Magnitude of uu: u=((1n)i+nj)((1n)i+nj)=(1n)2+n2\|u\| = \sqrt{((1 - n)i + nj) \cdot ((1 - n)i + nj)} = \sqrt{(1 - n)^2 + n^2} Simplifying: u=(12n+n2+n2)=12n+2n2\|u\| = \sqrt{(1 - 2n + n^2 + n^2)} = \sqrt{1 - 2n + 2n^2}

    • Magnitude of vv: v=(ij)(ij)=1+1=2\|v\| = \sqrt{(i - j) \cdot (i - j)} = \sqrt{1 + 1} = \sqrt{2}

  5. Angle Between uu and vv: Now, use the dot product and magnitudes to find cos(θ)\cos(\theta): cos(θ)=12n12n+2n22\cos(\theta) = \frac{1 - 2n}{\sqrt{1 - 2n + 2n^2} \cdot \sqrt{2}}

    This gives you the cosine of the angle between uu and vv. The angle θ\theta can then be calculated using: θ=cos1(12n2(12n+2n2))\theta = \cos^{-1}\left(\frac{1 - 2n}{\sqrt{2(1 - 2n + 2n^2)}}\right)

Let me know if you'd like to explore the solution in more detail or have any questions!


Here are 5 related questions for further exploration:

  1. How do you calculate the angle between vectors in 3D space?
  2. What is the geometric interpretation of the dot product?
  3. How do the values of nn affect the angle between uu and vv?
  4. What is the significance of perpendicular vectors in vector geometry?
  5. How does the cross product differ from the dot product when dealing with vectors?

Tip: Understanding the dot product helps in many areas of physics and geometry, especially when analyzing angles between vectors!

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Math Problem Analysis

Mathematical Concepts

Vector Algebra
Dot Product
Trigonometry

Formulas

Dot Product Formula: u ⋅ v = |u||v|cos(θ)
Magnitude of a Vector: |v| = √(v₁² + v₂²)
Cosine of Angle: cos(θ) = (u ⋅ v) / (|u||v|)

Theorems

Dot Product Theorem

Suitable Grade Level

Grades 11-12, College Level