Vectors

1. Scalar and Vector Quantities

• A scalar quantity has magnitude but no direction. Scalars add and subtract by the ordinary rules of algebra. Examples: mass, time, temperature, energy.
• A vector quantity has both magnitude and direction and obeys the rules of vector addition. Examples: displacement, velocity, acceleration, force, torque, weight.

A vector is represented graphically by a straight line with an arrowhead: the length of the line gives the magnitude and the arrow gives the direction.

Note: Having both magnitude and direction is necessary but not sufficient for a quantity to be a vector; it must also add by the vector (triangle/parallelogram) law. Electric current has magnitude and direction yet adds algebraically, so it is a scalar.

If $\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$, its magnitude is:

$$ \begin{aligned} |\vec{A}| &= A \ &= \sqrt{A_x^2 + A_y^2 + A_z^2} \end{aligned} $$

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2. Types of Vectors

• Unit vector: a vector of unit magnitude, used to show direction. For a vector $\vec{A}$, $\hat{A} = \dfrac{\vec{A}}{|\vec{A}|}$.
• Null (zero) vector: a vector of zero magnitude; its initial and terminal points coincide.
• Parallel vectors: vectors in the same direction (angle between them $0^{\circ}$).
• Anti-parallel vectors: vectors in exactly opposite directions (angle $180^{\circ}$).
• Equal vectors: same magnitude and same direction.
• Negative vectors: same magnitude but opposite direction.
• Coplanar vectors: vectors lying in the same plane.
• Co-initial vectors: vectors having the same starting point.
• Co-terminal vectors: vectors having the same end point.
• Orthogonal vectors: vectors perpendicular to each other ($90^{\circ}$).
• Collinear vectors: vectors acting along the same line (angle $0^{\circ}$ or $180^{\circ}$).

Example. Find the unit vector of $\vec{A} = 3\hat{i} + 4\hat{j} + 5\hat{k}$.

$$ \begin{aligned} |\vec{A}| &= \sqrt{3^2 + 4^2 + 5^2} \ &= \sqrt{50} \ &= 5\sqrt{2} \ \hat{A} &= \frac{3\hat{i} + 4\hat{j} + 5\hat{k}}{5\sqrt{2}} \end{aligned} $$

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3. Triangle Law of Vector Addition

If two vectors are represented, in magnitude and direction, by two sides of a triangle taken in order, then the third side taken in the opposite order represents their resultant. If $\vec{A}$ and $\vec{B}$ are the two sides, the resultant $\vec{R} = \vec{A} + \vec{B}$ closes the triangle.

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4. Parallelogram Law of Vector Addition

If two vectors are represented, in magnitude and direction, by the two adjacent sides of a parallelogram drawn from a point, then the diagonal through that point represents their resultant in both magnitude and direction.

Magnitude. Let $\vec{A}$ and $\vec{B}$ act at angle $\theta$. Producing $OP$ and dropping a perpendicular $TS$ from the tip $T$:

$$OT^2 = (OP + PS)^2 + (ST)^2$$

With $ST = B\sin\theta$ and $PS = B\cos\theta$:

$$ \begin{aligned} R^2 &= (A + B\cos\theta)^2 + (B\sin\theta)^2 \ &= A^2 + 2AB\cos\theta + B^2\cos^2\theta + B^2\sin^2\theta \ R^2 &= A^2 + B^2 + 2AB\cos\theta \quad\Rightarrow\quad \boxed{R = \sqrt{A^2 + B^2 + 2AB\cos\theta}} \end{aligned} $$

Direction. If the resultant makes angle $\alpha$ with $\vec{A}$:

$$\tan\alpha = \frac{B\sin\theta}{A + B\cos\theta}$$

(If $\vec{R}$ makes angle with $\vec{B}$ instead, $\tan\alpha = \dfrac{A\sin\theta}{B + A\cos\theta}$.)