A physical quantity that has only magnitude and no direction is called a scalar.

Displacement in two dimensions is represented by a vector drawn from the final position to the initial position.

The magnitude of a vector is always a non-negative real number.

Two vectors are equal if and only if they have the same magnitude and the same direction, regardless of their initial points.

Vector addition by the triangle method and by the parallelogram method always gives the same resultant.

If two non-zero vectors are perpendicular to each other, the magnitude of their resultant is the sum of their magnitudes.

A vector of given magnitude has a unique pair of rectangular components along any two fixed perpendicular directions.

The x-component of a vector can be greater in magnitude than the vector itself.

For any two vectors \(\vec{A}\) and \(\vec{B}\) in a plane, \(\vec{A}\cdot\vec{B}=0\) implies that the vectors are perpendicular.

The cross product of two non-parallel vectors lying in the same plane is a vector perpendicular to that plane.

In projectile motion on level ground (neglecting air resistance), the horizontal component of velocity remains constant throughout the motion.

In projectile motion, the vertical component of velocity is the same at any two points that are at the same height above the ground.

For a projectile launched with speed \(u\) at angle \(\theta\) above the horizontal, the time of flight on level ground is proportional to \(\cos\theta\).

For fixed launch speed on level ground, the horizontal range of a projectile is maximum at a projection angle of \(45^\circ\).

Two projectiles fired with the same speed at complementary angles \(\theta\) and \(90^\circ-\theta\) have the same maximum height.

The trajectory of a projectile under uniform gravity and without air resistance is a parabola when described in Cartesian coordinates.

In uniform circular motion, the speed and velocity of the particle both remain constant.

In uniform circular motion of radius \(r\) and speed \(v\), the acceleration is always directed radially inward and has magnitude \(v^{2}/r\).

A body moving in a circle with constant speed has zero tangential acceleration but non-zero normal (centripetal) acceleration.

If the velocity of object B relative to object A is \(\vec{v}_{BA}\), then the velocity of A relative to B is \(-\vec{v}_{BA}\).

In river-boat problems, if the boat is always steered perpendicular to the river bank, the shortest time to cross is achieved when the river flow speed is zero.

The relative velocity of rain with respect to a moving observer can be found by subtracting the observer’s velocity vector from the rain’s velocity vector.

For a projectile launched from ground and landing at a higher horizontal level, the equation \(R=\dfrac{u^{2}\sin 2\theta}{g}\) for range on level ground remains valid without modification.

In uniform circular motion, if the angular speed of a particle is doubled while the radius is halved, the magnitude of its centripetal acceleration remains unchanged.

For a projectile launched with speed \(u\) at angle \(\theta\), if its horizontal range on level ground is equal to its maximum height, then \(\tan\theta=8\).