JEE Physics Vectors and Calculus Notes Set 02

Scalars and Vectors

A scalar has magnitude only (mass, time, work). A vector has magnitude and direction and obeys the vector laws of addition (force, velocity, displacement).
Equal vectors have the same magnitude and direction. A unit vector \( \hat A=\vec A/A \) has magnitude 1. A null vector has zero magnitude.

Addition and Subtraction

Triangle law: place the tail of \( \vec{B} \) on the head of \( \vec{A} \); the resultant joins the free ends.
Magnitude of the resultant: \( R=\sqrt{A^2+B^2+2AB\cos\theta} \). Direction from \( \vec{A} \): \( \tan\alpha=\frac{B\sin\theta}{A+B\cos\theta} \).
Maximum resultant \( A+B \) at 0 degrees, minimum \( |A-B| \) at 180 degrees. For the difference, replace \( \cos\theta \) by \( -\cos\theta \).
Three vectors with a zero resultant always form a closed triangle, so they are coplanar.

Components

In a plane, \( \vec A=A_x\hat i+A_y\hat j \) with \( A_x=A\cos\theta,\ A_y=A\sin\theta \). In space, \( \vec A=A_x\hat i+A_y\hat j+A_z\hat k \) and \( A=\sqrt{A_x^2+A_y^2+A_z^2} \).
Direction cosines satisfy \( l^2+m^2+n^2=1 \). Add vectors by adding like components.

Scalar (Dot) Product

\( \vec A\cdot\vec B=AB\cos\theta=A_xB_x+A_yB_y+A_zB_z \). It is commutative and distributive. It is zero for perpendicular vectors.
Component of \( \vec{A} \) along \( \vec{B} \): \( A\cos\theta=\vec A\cdot\hat B \).

Vector (Cross) Product

\( \vec A\times\vec B=AB\sin\theta\,\hat n \), with direction given by the right-hand rule. It is not commutative: \( \vec A\times\vec B=-\vec B\times\vec A \). It is zero for parallel vectors.
In components it is the determinant of the rows (i, j, k), (Ax, Ay, Az), (Bx, By, Bz).
Area of a parallelogram = \( |\vec A\times\vec B| \) and of a triangle = \( \tfrac12|\vec A\times\vec B| \).

Triple Products

Scalar triple product \( \vec A\cdot(\vec B\times\vec C) \) equals the volume of the parallelepiped on the three vectors, and \( \vec A\cdot(\vec B\times\vec C)=\vec B\cdot(\vec C\times\vec A)=\vec C\cdot(\vec A\times\vec B) \). It is zero when the three vectors are coplanar.

Worked Examples

Example 1. Find the angle between \( \vec A=\hat i+\hat j \) and \( \vec B=\hat j+\hat k \).
Solution: \( \cos\theta=\frac{\vec A\cdot\vec B}{AB}=\frac{1}{\sqrt2\cdot\sqrt2}=\frac12 \), so \( \theta=60^\circ \).

Example 2. Find the area of the triangle formed by \( \vec A=2\hat i \) and \( \vec B=3\hat j \) as adjacent sides.
Solution: \( \vec A\times\vec B=6\hat k \), so the area is \( \tfrac12\times6=3 \) square units.