Physics vectors basic concepts and practice questions and objectives(mcqs) with solutions

Vectors
Scalar and Vector Quantities
Physical quantities are classified into two types:
 Scalar Quantities: Quantities that have only magnitude and no
direction. Examples: mass, distance, time, speed, volume, density,
pressure, work, energy, power, charge, electric current, temperature,
potential, specific heat, frequency.
 Vector Quantities: Quantities that have both magnitude and
direction. Examples: position, displacement, velocity, acceleration,
force, weight, momentum, impulse, electric field, magnetic field,
current density.

The Three Conditions for a Vector
A physical quantity is only a vector if it:
1. Has a specified direction.
Obeys the law of parallelogram addition: R = \sqrt{A^2 + B^2 + 2AB \cos
\theta}.
1. Obeys commutative addition: \vec{A} + \vec{B} = \vec{B} + \
vec{A}. Note: Having direction alone does not make a quantity a
vector (e.g., time, pressure, and current have direction but are not
vectors).

Types of Vectors
1. Polar Vectors: Have a starting point (displacement) or a point of
application (force).
1. Axial Vectors: Represent rotational effects. They act along the axis of
rotation, determined by the right-hand thumb rule. Examples: angular
velocity, torque, angular momentum.
1. Unit Vector: A vector having unit magnitude (|\hat{a}| = 1). It is used
to denote direction: \hat{A} = \vec{A} / |\vec{A}|.
1. Orthogonal Unit Vectors: The unit vectors along the X, Y, and Z axes
are written as \hat{i}, \hat{j}, and \hat{k}.
1. Null Vector: A vector with zero magnitude and no specific direction.

Vector Algebra and Representation
 Translation: A vector remains unchanged if displaced parallel to itself.
 Rotation: A vector changes if rotated by an angle other than 360^\
circ.
 Multiplication by Scalar: Multiplying \vec{A} by a scalar n results in
a new vector with magnitude n|\vec{A}|.

,  Dot Product (Scalar Product): \vec{A} \cdot \vec{B} = AB \cos \
theta = A_x B_x + A_y B_y + A_z B_z.
 Cross Product (Vector Product): \vec{A} \times \vec{B} = (AB \
sin \theta) \hat{n}.


50 Practice Questions on Vectors
Level 1: Basics and Magnitude
1. Find the magnitude of \vec{A} = 3\hat{i} - 4\hat{j} + 5\hat{k}.
Solution: |\vec{A}| = \sqrt{3^2 + (-4)^2 + 5^2} = \sqrt{9 + 16 + 25} = \sqrt{50} = 5\sqrt{2}.
2. A vector has magnitude 10. If its x-component is 6, find the y-component.
Solution: 10 = \sqrt{6^2 + y^2} \rightarrow 100 = 36 + y^2 \rightarrow y = \sqrt{64} = 8.
3. Find the unit vector of \vec{B} = \hat{i} + \hat{j}.
Solution: |\vec{B}| = \sqrt{1^2 + 1^2} = \sqrt{2}. \hat{b} = \frac{1}{\sqrt{2}}\hat{i} + \frac{1}
{\sqrt{2}}\hat{j}.
4. Find 2\vec{A} - \vec{B} if \vec{A} = (1, 2) and \vec{B} = (3, 0).
Solution: 2(1, 2) - (3, 0) = (2, 4) - (3, 0) = (-1, 4).
5. Are mass and pressure vectors or scalars?
Solution: Scalars.
6. Calculate the direction cosines of \vec{r} = 2\hat{i} + 3\hat{j} + 6\hat{k}.
Solution: |\vec{r}| = 7. l = 2/7, m = 3/7, n = 6/7.
7. If |\vec{A} + \vec{B}| = |\vec{A} - \vec{B}|, find the angle between \vec{A} and \vec{B}.
Solution: 90^\circ. Dot product is zero.
8. Define a Null Vector.
Solution: Magnitude is zero; no specific direction.
9. Can a force vector be added to a velocity vector?
Solution: No, only vectors of the same nature can be added.
10. Calculate |\vec{A} + \vec{B}| if A=3, B=4 and they are perpendicular.
Solution: \sqrt{3^2 + 4^2} = 5.

Level 2: Operations and Products
11. Find \vec{A} \cdot \vec{B} if \vec{A} = \hat{i} + 2\hat{j} and \vec{B} = 3\hat{i} - \hat{j}.
Solution: (1)(3) + (2)(-1) = 3 - 2 = 1.
12. Find the angle between \vec{A} = \hat{i} and \vec{B} = \hat{j}.
Solution: 90^\circ (Orthogonal unit vectors).
13. Calculate \vec{A} \times \vec{B} for \vec{A} = \hat{i} and \vec{B} = \hat{j}.
Solution: \hat{k}.
14. If \vec{A} and \vec{B} are parallel, what is \vec{A} \times \vec{B}?
Solution: Zero vector.