Lesson 5 Vector Projections

5.1 Lecture Content

Video: Vectors, Dot Products, and Projections

5.2 Lecture Notes

5.2.1 Vector Operations

Addition and Scalar Multiplication of Vectors

  • Vector Addition: If u = ⟨u₁, u₂, …, uₙ⟩ and v = ⟨v₁, v₂, …, vₙ⟩, then
    u + v = ⟨u₁ + v₁, u₂ + v₂, …, uₙ + vₙ⟩.

  • Scalar Multiplication: For scalar c,
    c·v = ⟨c·v₁, c·v₂, …, c·vₙ⟩.

  • Example: Let u = ⟨1, 2, 3⟩ and v = ⟨4, -1, 2⟩ Then: \[ \mathbf{u} + \mathbf{v} = \langle 1 + 4,\ 2 + (-1),\ 3 + 2 \rangle = \langle 5,\ 1,\ 5 \rangle \]

  • Example: Let v = ⟨2, -3, 4> and and scalar c = 3. Then: \[ 3 \cdot \mathbf{v} = \langle 3 \cdot 2,\ 3 \cdot (-3),\ 3 \cdot 4 \rangle = \langle 6,\ -9,\ 12 \rangle \]

These operations preserve vector direction (in the case of addition) or scale the magnitude (in scalar multiplication).

5.2.2 Standard Inner Product (Dot Product)

The standard inner product (dot product) of vectors u and v in ℝⁿ is: \[ \mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + \cdots + u_n v_n \]

  • Example: Let u = ⟨1, 2, 3⟩ and v = ⟨4, -1, 2⟩ Then: \[ \mathbf{u} \cdot \mathbf{v} = 1 \cdot 4 + 2 \cdot (-1) + 3 \cdot 2 = 4 - 2 + 6 = 8 \]
  • Returns a scalar.
  • Measures how much one vector goes in the direction of another.

5.2.3 Euclidean Norm of a Vector

The Euclidean norm (or magnitude/length) of a vector v is:

\[ \| \mathbf{v} \| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2} \]

  • Example: Let v = ⟨2, -3, 6⟩. Then the 2-norm is: \[ \| \mathbf{v} \| = \sqrt{2^2 + (-3)^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]
  • Denoted as |v|.
  • Also called the 2-norm.

5.2.4 Cosine of the Angle Between Two Vectors

Given non-zero vectors u and v, the cosine of the angle θ between them is:

\[ \cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|} \]

  • Example: Let u = ⟨1, 2⟩ and v = ⟨3, 4⟩
    1. Compute dot product:
      \[ \mathbf{u} \cdot \mathbf{v} = 1 \cdot 3 + 2 \cdot 4 = 3 + 8 = 11 \]
    2. Compute norms:
      \[ \|\mathbf{u}\| = \sqrt{1^2 + 2^2} = \sqrt{5}, \quad \|\mathbf{v}\| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \]
    3. Use the cosine formula:
      \[ \cos(\theta) = \frac{11}{\sqrt{5} \cdot 5} = \frac{11}{5\sqrt{5}} = \frac{\sqrt{5}}{5} \cdot \frac{11}{5} \]
    4. Find angle:
      \[ \theta = \cos^{-1}\left( \frac{11}{5\sqrt{5}} \right) \approx 18.4^\circ \]
  • If cos(θ) = 0, the vectors are orthogonal (perpendicular).
  • If cos(θ) > 0, angle < 90° (acute).
  • If cos(θ) < 0, angle > 90° (obtuse).

5.2.5 Vector Projection Formula

The projection of vector u onto vector v is:

\[ \mathrm{proj}_{\mathbf{v}} \mathbf{u} = \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \right) \mathbf{v} \]

  • Example: Project u = ⟨2, 3⟩ onto v = ⟨1, 4⟩.
    1. Dot product:
      \[ \mathbf{u} \cdot \mathbf{v} = 2 \cdot 1 + 3 \cdot 4 = 2 + 12 = 14 \]
    2. Norm squared:
      \[ \|\mathbf{v}\|^2 = 1^2 + 4^2 = 1 + 16 = 17 \]
    3. Projection:
      \[ \mathrm{proj}_{\mathbf{v}} \mathbf{u} = \left( \frac{14}{17} \right) \mathbf{v} = \left( \frac{14}{17} \right) \langle 1,\ 4 \rangle = \left\langle \frac{14}{17},\ \frac{56}{17} \right\rangle \]
  • Gives the component of u in the direction of v.

5.2.6 Geometric Interpretations

  • Addition: Geometrically, vector addition corresponds to placing the tail of v at the head of u and drawing the resulting vector from the tail of u to the head of v.

  • Dot Product: Measures alignment. A large positive value means the vectors point in similar directions.

  • Norm: The “length” of the vector in Euclidean space.

  • Projection: Drops a perpendicular from u onto v; the projection lies along v.

  • Angle and Cosine: Relates direction and orthogonality.

5.3 Examples

Video: Solutions

  1. Consider the following vectors:
    \[ \vec{v} = \langle -11, -2 \rangle, \quad \vec{z} = \langle 6, 5 \rangle \]

Compute:
\[ \vec{v} + \vec{z} \]

  1. Consider the following vector:
    \[ \vec{v} = \langle 8, -6 \rangle \]

Find: \[ \|\vec{v}\| = \ ? \]

  1. The equation \(ax + by = c\) corresponds to the vector \(\langle a, b, c \rangle\).

    1. To what equation will \(\langle 3, -1, 4 \rangle\) correspond?
    2. To what equation will \(\langle -5, 3, -1 \rangle\) correspond?
    3. To what equation will \(\langle 3, -1, 4 \rangle + \langle -5, 3, -1 \rangle\) correspond?
    4. What should the corresponding vector be?

  2. Let: \[ \vec{u} = [4, 1] \quad \vec{v} = [4, 1] \]

    1. Draw \(\vec{u}\) in standard position (tail at the origin).
    2. Draw \(\vec{v}\) with its tail at the tip of \(\vec{u}\).
    3. Compute \(\vec{u} + \vec{v}\).
    4. Draw \(\vec{u} + \vec{v}\) from the tail of \(\vec{u}\) to the tip of \(\vec{v}\).

  3. Use the dot product to determine if the vectors are perpendicular/ orthogonal.

  1. \[ \vec{u} = \begin{bmatrix} 7 \\ 1 \\ 1 \end{bmatrix}, \quad \vec{v} = \begin{bmatrix} 4 \\ 2 \\ 5 \end{bmatrix} \]

  2. \[ \vec{u} = \begin{bmatrix} -4 \\ -4 \\ -8 \end{bmatrix}, \quad \vec{v} = \begin{bmatrix} -1 \\ -1 \\ 1 \end{bmatrix} \] \(\vec{u} \cdot \vec{v} =\) ?

  1. Compute the dot product:
    \[ \begin{bmatrix} -10 \\ 4 \\ -4 \\ -2 \end{bmatrix} \cdot \begin{bmatrix} 5 \\ 7 \\ -3 \\ 7 \end{bmatrix} \]

  2. Find the angle between the vectors:
    \[ \begin{bmatrix} -6 \\ 8 \\ -8 \end{bmatrix} \quad \text{and} \quad \begin{bmatrix} 8 \\ -9 \\ 6 \end{bmatrix} \]
    Give your answer in radians (rounded to two decimal places).

  3. Write all answers with radicals and/or fractions as needed. Do not use decimals.

    1. Find the magnitude (norm) of
      \[ \vec{u} = [-6, 1] \]
      Find a unit vector in the direction of \(\vec{u}\).

    2. Find the magnitude (norm) of
      \[ \vec{v} = \begin{bmatrix} -3 \\ 4 \\ 0 \end{bmatrix} \]
      Find a unit vector in the direction of \(\vec{v}\).

    3. Find the magnitude (norm) of
      \[ \vec{w} = \begin{bmatrix} 6 \\ 1 \\ 5 \\ 3 \end{bmatrix} \]
      Find a unit vector in the direction of \(\vec{w}\).

5.4 Practice Problems

  1. Draw the vectors \(A = \begin{bmatrix} 1 \\ 2 \end{bmatrix}\) and \(B = \begin{bmatrix} -1 \\ -1 \end{bmatrix}\). Draw:
    • \(A + B\),
    • \(B + A\),
    • \(A - B\),
    • \(B - A\),
    • \(2A\), and
    • \(-4B\).
  2. Compute the standard inner products and norms of \(A = \begin{bmatrix} -1 \\ 3 \end{bmatrix}\) and \(B = \begin{bmatrix} 2 \\ 2 \end{bmatrix}\). What is the angle between them?
  3. Express \(v = [5, -1]\) as a sum of orthogonal vectors such that one has the same direction as \(u = [4, 2]\).
  4. Find the angle between:
    • \(x = \begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix}\) and \(y = \begin{bmatrix} 2 \\ 0 \\ -3 \end{bmatrix}\)
    • \(x = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}\) and \(y = \begin{bmatrix} 3 \\ -2 \\ 1 \end{bmatrix}\)
  5. Find the projection and orthogonal projection of \(u\) onto \(v\):
    • \(u = \begin{bmatrix} 4 \\ 3 \end{bmatrix}\), \(v = \begin{bmatrix} 2 \\ 8 \end{bmatrix}\)
    • \(u = \begin{bmatrix} 3 \\ 5 \end{bmatrix}\), \(v = \begin{bmatrix} 6 \\ 2 \end{bmatrix}\)

5.5 Self-Assessment

Try to solve the following within 20 minutes.

  1. Draw \(A = \begin{bmatrix} -1 \\ 3 \end{bmatrix}\) and \(B = \begin{bmatrix} 2 \\ 2 \end{bmatrix}\). Then draw:
    • \(A + B\),
    • \(A - B\),
    • \(B - A\),
    • \(3A\),
    • \(-0.5B\)
  2. Find the angle between: \(x = \begin{bmatrix} 1 \\ 2 \end{bmatrix}\) and \(y = \begin{bmatrix} 2 \\ 1 \end{bmatrix}\)
  3. Find the projection and orthogonal projection of \(u\) onto \(v\): \(u = \begin{bmatrix} -2 \\ 5 \end{bmatrix}\), \(v = \begin{bmatrix} 6 \\ 5 \end{bmatrix}\)
  4. Express \(v = [8, -3, -3]\) as a sum of orthogonal vectors with one in the direction of \(u = [2, 3, 2]\)

5.6 Lesson Checklist

This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an \(X\) in the appropriate box beside the skill below.

\[\begin{align*} &\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\ &\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\ &\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} \end{align*}\]

Skill D CON COM
Determine the inner product of two vectors
Determine the Euclidean norm of a vector
Find the cosine of the angle between two vectors
Find a vector projection and an orthogonal projection
Decompose a vector into orthogonal vectors