Vectors - Class 11 Physics

Definition: A vector is a mathematical quantity that has both magnitude (size) and direction. Unlike scalars, which only have magnitude, vectors are used to represent physical quantities like displacement, velocity, force, and acceleration.

Vectors are typically represented by an arrow, where:

• The length of the arrow represents the magnitude
• The arrowhead indicates the direction

Examples of Vector Quantities:

• Displacement
• Velocity
• Acceleration
• Force
• Momentum
• Electric field

Arrow Notation: \(\vec{a}\) or \(\overrightarrow{AB}\)

Component Form (2D):

\[\vec{a} = a_x\hat{i} + a_y\hat{j}\]

or

\[\vec{a} = (a_x, a_y)\]

Component Form (3D):

\[\vec{a} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}\]

or

\[\vec{a} = (a_x, a_y, a_z)\]

Magnitude:

\[|\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}\]

Standard Unit Vectors:

• \(\hat{i} = (1, 0, 0)\) - unit vector along x-axis
• \(\hat{j} = (0, 1, 0)\) - unit vector along y-axis
• \(\hat{k} = (0, 0, 1)\) - unit vector along z-axis

1. Zero Vector (\(\vec{0}\)): Magnitude = 0, no specific direction

2. Unit Vector (\(\hat{a}\)): Magnitude = 1

\[\hat{a} = \frac{\vec{a}}{|\vec{a}|}\]

3. Position Vector: Vector from origin to a point P(x, y, z)

\[\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}\]

4. Equal Vectors: Same magnitude and direction

5. Negative Vector (\(-\vec{a}\)): Same magnitude, opposite direction

6. Collinear Vectors: Vectors along the same line (parallel or anti-parallel)

7. Coplanar Vectors: Vectors lying in the same plane

Triangle Law: Place vectors head-to-tail. The resultant is from tail of first to head of last.

Parallelogram Law: The resultant is the diagonal of the parallelogram formed by the two vectors.

Component Method:

\[\vec{R} = \vec{a} + \vec{b} = (a_x + b_x)\hat{i} + (a_y + b_y)\hat{j} + (a_z + b_z)\hat{k}\]

Properties:

• Commutative: \(\vec{a} + \vec{b} = \vec{b} + \vec{a}\)
• Associative: \((\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})\)
• Additive Identity: \(\vec{a} + \vec{0} = \vec{a}\)

Vector subtraction is defined as:

\[\vec{a} - \vec{b} = \vec{a} + (-\vec{b})\]

Component Method:

\[\vec{a} - \vec{b} = (a_x - b_x)\hat{i} + (a_y - b_y)\hat{j} + (a_z - b_z)\hat{k}\]

Multiplying a vector by a scalar \(k\):

\[k\vec{a} = ka_x\hat{i} + ka_y\hat{j} + ka_z\hat{k}\]

Properties:

• If \(k > 0\): Same direction, magnitude scaled by \(|k|\)
• If \(k < 0\): Opposite direction, magnitude scaled by \(|k|\)
• If \(k = 0\): Results in zero vector

Magnitude: \(|k\vec{a}| = |k||\vec{a}|\)

Definition:

\[\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta\]

where \(\theta\) is the angle between the vectors.

Component Form:

\[\vec{a} \cdot \vec{b} = a_xb_x + a_yb_y + a_zb_z\]

Result: Scalar (number)

Key Properties:

• If \(\vec{a} \cdot \vec{b} = 0\), vectors are perpendicular
• \(\vec{a} \cdot \vec{a} = |\vec{a}|^2\)
• \(\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1\)
• \(\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0\)

Application: Work done = \(\vec{F} \cdot \vec{d}\)

Magnitude:

\[|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta\]

Result: Vector perpendicular to both \(\vec{a}\) and \(\vec{b}\)

Direction: Right-hand rule

Determinant Method:

\[\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}\]

Key Properties:

• \(\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}\) (anti-commutative)
• \(\vec{a} \times \vec{a} = \vec{0}\)
• \(\hat{i} \times \hat{j} = \hat{k}\), \(\hat{j} \times \hat{k} = \hat{i}\), \(\hat{k} \times \hat{i} = \hat{j}\)

Applications:

• Torque: \(\vec{\tau} = \vec{r} \times \vec{F}\)
• Area of parallelogram: \(|\vec{a} \times \vec{b}|\)

For a vector \(\vec{a} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}\):

Direction Cosines:

• \(\cos\alpha = \frac{a_x}{|\vec{a}|}\) (angle with x-axis)
• \(\cos\beta = \frac{a_y}{|\vec{a}|}\) (angle with y-axis)
• \(\cos\gamma = \frac{a_z}{|\vec{a}|}\) (angle with z-axis)

Identity:

\[\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\]