Vector projection is a fundamental operation in linear algebra that finds the component of one vector in the direction of another vector. This calculation is essential in physics for analyzing forces, in computer graphics for lighting calculations, and in engineering for determining directional components of vectors.
\[ \text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{\vec{b} \cdot \vec{b}} \vec{b} \]
\[ \text{comp}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \]
| Symbol | Definition | Units/Range |
|---|---|---|
| \(\vec{a}\) | The vector being projected (input vector) | Any real numbers for components |
| \(\vec{b}\) | The vector onto which we project (target vector) | Non-zero vector (any real numbers) |
| \(\vec{a} \cdot \vec{b}\) | Dot product of vectors \(\vec{a}\) and \(\vec{b}\) | Scalar value (can be positive, negative, or zero) |
| \(|\vec{b}|\) | Magnitude (length) of vector \(\vec{b}\) | Always positive (> 0) |
| \(\vec{b} \cdot \vec{b}\) | Dot product of \(\vec{b}\) with itself, equals \(|\vec{b}|^2\) | Always positive (> 0) |
Theoretical Foundation: Vector projection geometrically represents the "shadow" that vector \(\vec{a}\) casts onto vector \(\vec{b}\) when light shines perpendicular to \(\vec{b}\).
Mathematical Basis: The projection formula \(\text{proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{\vec{b} \cdot \vec{b}} \vec{b}\) ensures that:
Given: Find the projection of vector \(\vec{a} = (4, 3)\) onto vector \(\vec{b} = (2, 1)\)
Physical Interpretation: Imagine vector \(\vec{a}\) represents a force of 4 units in the x-direction and 3 units in the y-direction. We want to find how much of this force acts in the direction of vector \(\vec{b}\).
Formula: \(\vec{a} \cdot \vec{b} = a_x \cdot b_x + a_y \cdot b_y\)
Calculation:
\(\vec{a} \cdot \vec{b} = (4)(2) + (3)(1) = 8 + 3 = 11\)
Interpretation: The dot product measures how much the vectors "agree" in direction. A positive value indicates they point in generally the same direction.
Formula: \(\vec{b} \cdot \vec{b} = b_x^2 + b_y^2 = |\vec{b}|^2\)
Calculation:
\(\vec{b} \cdot \vec{b} = (2)^2 + (1)^2 = 4 + 1 = 5\)
Note: We use \(\vec{b} \cdot \vec{b}\) instead of \(|\vec{b}|\) to avoid taking square roots, which simplifies calculations.
Vector Projection Calculation:
\(\text{proj}_{\vec{b}}\vec{a} = \frac{11}{5} \times (2, 1) = 2.2 \times (2, 1) = (4.4, 2.2)\)
Scalar Projection:
\(\text{comp}_{\vec{b}}\vec{a} = \frac{11}{\sqrt{5}} = \frac{11}{2.236} \approx 4.919\)
Verification: Check that the projection is parallel to \(\vec{b}\):
\(\frac{4.4}{2} = 2.2\) and \(\frac{2.2}{1} = 2.2\) ✓ (Both ratios are equal)
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