Section 1.3: Vectors

This section focuses on vectors and their operations. We learn how to represent vectors, perform addition and subtraction, calculate components, and apply these concepts in physics problems.

Vector Representation:

A vector has magnitude and direction: \( \vec{A} = A_x \hat{i} + A_y \hat{j} \)

Vector Addition:

Graphical method: tip-to-tail method or parallelogram method.

Component method: \( \vec{R} = \sum \vec{A}_i = (\sum A_{ix}) \hat{i} + (\sum A_{iy}) \hat{j} \)

Vector Subtraction:

\( \vec{A} - \vec{B} = \vec{A} + (-\vec{B}) \)

Components: \( (A_x - B_x) \hat{i} + (A_y - B_y) \hat{j} \)

Example 1

Vector \( \vec{A} = 3\hat{i} + 4\hat{j} \), Vector \( \vec{B} = 1\hat{i} + 2\hat{j} \). Find \( \vec{R} = \vec{A} + \vec{B} \).

\( \vec{R} = (3+1)\hat{i} + (4+2)\hat{j} = 4\hat{i} + 6\hat{j} \)

Magnitude: \( |\vec{R}| = \sqrt{4^2 + 6^2} = \sqrt{52} \approx 7.21 \)

Direction: \( \theta = \tan^{-1}(6/4) \approx 56.3^\circ \) above x-axis

Practice Problems

  1. Vectors \( \vec{A} = 5\hat{i} + 2\hat{j} \), \( \vec{B} = -3\hat{i} + 4\hat{j} \). Find \( \vec{A} + \vec{B} \).
  2. Subtract vectors: \( \vec{C} = 6\hat{i} + 3\hat{j} \), \( \vec{D} = 2\hat{i} + 7\hat{j} \).
  3. Find the magnitude and direction of \( \vec{E} = 4\hat{i} - 3\hat{j} \).
  4. Vector components: \( \vec{F} = 10 \text{ at } 30^\circ \) above x-axis. Find \( F_x, F_y \).
  5. Given vectors \( \vec{G} = 3\hat{i} + 4\hat{j} \), \( \vec{H} = -1\hat{i} + 2\hat{j} \). Compute \( \vec{G} - \vec{H} \).
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