Vectors and Advanced Mathematics Topics

a) Parallelogram Law of Vector Addition

Statement: If two vectors acting simultaneously at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from a point, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through that point.

Explanation: Let two vectors $\vec{P}$ and $\vec{Q}$ act at a point O, inclined at an angle $\theta$. They are represented by the sides OA and OB of a parallelogram OACB. The resultant $\vec{R}$ is represented by the diagonal OC.

Derivation of magnitude:

Drop a perpendicular from C on the extended line of OA, meeting it at N. In right-angled triangle ONC:

• $ON = OA + AN = P + Q\cos\theta$
• $CN = Q\sin\theta$
• $OC^2 = ON^2 + CN^2$

$R^2 = (P + Q\cos\theta)^2 + (Q\sin\theta)^2$ $R^2 = P^2 + 2PQ\cos\theta + Q^2\cos^2\theta + Q^2\sin^2\theta$ $R^2 = P^2 + Q^2 + 2PQ\cos\theta$

$\boxed{R = \sqrt{P^2 + Q^2 + 2PQ\cos\theta}}$

Derivation of direction:

Let the resultant make an angle $\alpha$ with $\vec{P}$. In triangle ONC:

$\tan\alpha = \frac{CN}{ON} = \frac{Q\sin\theta}{P + Q\cos\theta}$

$\boxed{\alpha = \tan^{-1}\left(\frac{Q\sin\theta}{P + Q\cos\theta}\right)}$

Special cases:

• When $\theta = 0°$: $R = P + Q$ (maximum)
• When $\theta = 180°$: $R = P - Q$ (minimum)
• When $\theta = 90°$: $R = \sqrt{P^2 + Q^2}$

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b) Difference between Scalar Product and Vector Product

Examples:

• Scalar product: Work done $W = \vec{F}\cdot\vec{d} = Fd\cos\theta$ (a scalar); Power $P = \vec{F}\cdot\vec{v}$.
• Vector product: Torque $\vec{\tau} = \vec{r}\times\vec{F}$; angular momentum $\vec{L} = \vec{r}\times\vec{p}$; magnetic force $\vec{F} = q(\vec{v}\times\vec{B})$.