Mechanics I: The Dot Product and Its Geometrical Applications

The Dot Product

Definition and Basic Properties

The dot product (also known as the scalar product) is a fundamental operation in vector algebra, widely used in physics and engineering to relate vectors to scalar quantities. Given two vectors a and b in two dimensions:

• Definition: For a = \langle a_1, a_2 \rangle and b = \langle b_1, b_2 \rangle, the dot product is defined as:

$a \cdot b = a_1 b_1 + a_2 b_2$

• Commutativity: The dot product is commutative, meaning $a \cdot b = b \cdot a$.

• Distributivity: The dot product distributes over vector addition and subtraction:

$a \cdot (b + c) = a \cdot b + a \cdot c$ $a \cdot (b - c) = a \cdot b - a \cdot c$ $(a + b) \cdot c = a \cdot c + b \cdot c$ $(a - b) \cdot c = a \cdot c - b \cdot c$

• Example: For a = \langle 1, 2 \rangle, b = \langle 9, -7 \rangle, c = \langle 4, 2 \rangle:

$a \cdot (2b + 3c) = 1 \times 30 + 2 \times (-8) = 14$

Alternatively, distributing:

$a \cdot (2b + 3c) = a \cdot (2b) + a \cdot (3c) = 1 \times 18 + 2 \times (-14) + 1 \times 12 + 2 \times 6 = 14$

Geometrical Interpretation of the Dot Product

Dot Product and Magnitude of a Vector

The dot product of a vector with itself yields the square of its magnitude:

• Formula: $a \cdot a = a_1^2 + a_2^2$

• Magnitude: $\|a\| = \sqrt{a_1^2 + a_2^2}$

• Thus, $\|a\| = \sqrt{a \cdot a}$ and $a \cdot a = \|a\|^2$

Dot Product and the Angle Between Vectors

The dot product relates to the angle between two vectors, providing a method to calculate the angle:

• Given vectors a and b with magnitudes $\|a\|$ and $\|b\|$, and angle $\theta$ between them:

$a \cdot b = \|a\| \|b\| \cos \theta$

• Solving for the angle:

$\theta = \arccos \left( \frac{a \cdot b}{\|a\| \|b\|} \right)$

• Domain: $-1 \leq \frac{a \cdot b}{\|a\| \|b\|} \leq 1$ ensures a real angle $0^\circ \leq \theta \leq 180^\circ$ (or $0 \leq \theta \leq \pi$ radians).

• Conversion: $1^\circ = \frac{\pi}{180}$ rad, $1$ rad $= \frac{180^\circ}{\pi}$

• Example: For a = \langle 5, 1 \rangle, b = \langle 2, 4 \rangle:

$\|a\| = \sqrt{5^2 + 1^2} = \sqrt{26}$ $\|b\| = \sqrt{2^2 + 4^2} = \sqrt{20}$ $a \cdot b = 5 \times 2 + 1 \times 4 = 14$ $\theta = \arccos \left( \frac{14}{\sqrt{26} \sqrt{20}} \right) \approx 52.13^\circ$ $\theta = 52.13 \times \frac{\pi}{180} \approx 0.91$ rad

Perpendicular Vectors

Vectors are perpendicular if the angle between them is $90^\circ$ (or $\frac{\pi}{2}$ radians). The dot product provides a test for perpendicularity:

• Criterion: $a \cdot b = 0$ if and only if a and b are perpendicular.

• Example: For a = \langle 1, 2 \rangle, b = \langle 3, x \rangle, if $a \cdot b = 0$ then $1 \times 3 + 2x = 0 \Rightarrow x = -\frac{3}{2}$

Geometrical Theorems by Vector Methods

Law of Cosines via Vector Algebra

Vector methods can be used to prove classical geometric theorems, such as the law of cosines:

• For triangle OAB with vectors a and b from O to A and O to B, the vector from B to A is a - b.

• Magnitude squared:

$\|a - b\|^2 = (a - b) \cdot (a - b) = a \cdot a + b \cdot b - 2(a \cdot b)$ $= \|a\|^2 + \|b\|^2 - 2\|a\|\|b\| \cos \theta$

• This is the law of cosines:

$AB^2 = OA^2 + OB^2 - 2 \cdot OA \cdot OB \cos \theta$

• Example: If $OA = 5$ units, $OB = 3$ units, $\theta = 30^\circ$:

$AB^2 = 5^2 + 3^2 - 2 \times 5 \times 3 \cos 30^\circ = 8.019$ units$^2$ $AB = \sqrt{8.019} \approx 2.83$ units

Parallelogram Properties (Exercise)

When two vectors have equal magnitude, their geometric arrangement in a parallelogram yields important properties:

• If $\|a\| = \|b\|$, then:

• (i) The diagonal from O to C bisects the angle $\theta$.

• (ii) The diagonals OC and BA bisect each other at $90^\circ$.

Additional info: These properties are foundational in vector geometry and are often used in physics to analyze forces and motion in two dimensions.