Vector Multiplication, Magnitude, and Component Addition

Scalar multiplication and vector scaling

A scalar multiplies a vector by scaling both components at the same time: if v = \langle x, y \rangle, then for a scalar c, c\u0304v = \langle c x, c y \rangle.
Example from transcript idea: if a vector has components (x\,=\,10) and (y\,=\,-6), then multiplying by 3 yields (3\u007e v = \langle 30, -18 \rangle).
Key point: scalar multiplication preserves the direction if c > 0, reverses it if c < 0, and scales the length by (|c|).

Vectors can be represented in multiple ways:
- By magnitude and direction (polar form): r, \theta \u007e.
- By Cartesian components (x, y): \u007f x, y \u007e.
The magnitude is the length of the vector, often denoted ||\u007f v ||. In Cartesian form, magnitude is computed from components.
The transcript notes that sometimes the magnitude form is convenient, but components are often more convenient for addition and further operations.

If a vector has magnitude (r) and direction angle (\theta) from the +x axis, its components are:
x = r \cos \theta, \quad y = r \sin \theta.
Example from transcript:
- Given (r = 10) km and (\theta = 36.876^{\circ}), compute
  x = 10 \cos(36.876^{\circ}) \approx 8 \text{ km}, \quad y = 10 \sin(36.876^{\circ}) \approx 6 \text{ km}.
- So the second leg has components approximately (\mathbf{v}_2 \approx \langle 8, 6 \rangle) km.
This conversion explains how to turn a single-direction, single-magnitude vector into two orthogonal components that can be added with another vector.
Note on angles: ensure angle is measured with respect to the same reference axis (usually the +x axis) and in compatible units (degrees or radians).

Given leg 1 (start vector) with components: (\mathbf{v}_1 = \langle 4, 6 \rangle) km.
Given leg 2 with magnitude 10 km at (\theta = 36.876^{\circ}).
- Convert to components: (\mathbf{v}_2 = \langle 8, 6 \rangle) km (approximately).
Net displacement is the sum of the legs:
\mathbf{v}{\text{net}} = \mathbf{v}1 + \mathbf{v}_2 = \langle 4+8, 6+6 \rangle = \langle 12, 12 \rangle \text{ km}.
Interpretation from the transcript:
- The y-component increases from 6 to 12 ("we went up six" then to 12).
- The x-component increases from 4 to 12 (resulting in a total of 12 for the x-component as well).
Net distance vs net displacement:
- The magnitude of the net displacement is the net distance along the straight-line from start to end.
- The direction is the angle of (\mathbf{v}_{\text{net}}) with respect to the +x axis.

Magnitude:
||\mathbf{v}_{\text{net}}|| = \sqrt{(12)^2 + (12)^2} = \sqrt{288} = 12\sqrt{2} \approx 16.97 \text{ km}.
Direction (angle from +x):
\phi = \tan^{-1}\left( \frac{v{\text{net},y}}{v{\text{net},x}} \right) = \tan^{-1}\left( \frac{12}{12} \right) = 45^{\circ}.
Result: net displacement is ( \mathbf{v}_{\text{net}} = \langle 12, 12 \rangle ) km with magnitude (12\sqrt{2}) km and direction (45^{\circ}) above the +x axis.

Purpose: to perform physics in more than one dimension by resolving vectors into components.
Benefits:
- Enables straightforward addition of displacements along different directions.
- Allows computation of net distance (magnitude) and net direction (angle).
- Provides a foundation for analyzing two-dimensional problems in mechanics (e.g., projectile motion, forces in the plane).
Conceptual takeaway: any path broken into straight segments can be analyzed by summing its vector legs component-wise to obtain the net displacement.

Scalar multiplication of a vector:
c \cdot \langle x, y \rangle = \langle c x, c y \rangle.
Cartesian representation from polar form:
\langle x, y \rangle = \langle r \cos \theta, r \sin \theta \rangle.
Polar from Cartesian:
r = ||\langle x, y \rangle|| = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1}\left( \frac{y}{x} \right).
Vector addition (2D):
\mathbf{v}1 + \mathbf{v}2 = \langle v{1x} + v{2x}, v{1y} + v{2y} \rangle.
Magnitude of a vector:
||\mathbf{v}|| = \sqrt{vx^2 + vy^2}.
Example constants used in the transcript: (r = 10\text{ km}, \theta = 36.876^{\circ}) gives approximately (\mathbf{v}_2 \approx \langle 8, 6 \rangle) km.

Always check units for consistency (e.g., all in kilometers, all angles in degrees or radians consistently).
When combining multiple legs, resolve each leg to components before summing; this reduces errors in multi-step problems.
The same method generalizes to three dimensions by adding a z-component if needed, using ( \mathbf{v} = \langle vx, vy, v_z \rangle ).