Is acceleration a vector or scalar?

In our study of physics, we deal with various physical quantities. Acceleration is one such physical quantity that is defined as the rate of change of velocity of an object with respect to time.

Our current topic of discussion is whether acceleration is a vector or scalar.

Again we know that physical quantities can be divided into two categories depending on whether they have only magnitude or they have both magnitude and direction. Scalar quantities have only magnitude. Vector quantities have both magnitude and direction. 

See the article on vectors and scalars in physics.

Vector quantities are often used to describe forces such as gravity, wind, or electric fields. Scalar quantities are usually associated with motion and include speed, distance, and mass.

Why acceleration is a vector quantity?

Let us look at whether acceleration has a direction with the help of two examples 

Speeding up and slowing down the car

When an object accelerates in the same direction as it moves (as shown in the figure), it is said to have a positive acceleration.

When an object experiences negative acceleration (slowing down), the acceleration is in the opposite direction of the object’s velocity.

A ball is thrown in an upward direction

Consider a ball being thrown into the air with some initial velocity $u$ as shown below in the figure. The ball accelerates at a constant rate of $g = 9.8 m/s^2$ [down] due to gravity.

When the ball is moving upward, the acceleration is in the other direction, causing the ball to slow down. Gravity continues to act on the ball even as it slows to a velocity of 0 m/s.

The ball then begins to fall since gravity continues to work on it, but now the motion and acceleration are in the same direction, causing the ball to accelerate.

From these two examples, we can clearly see that acceleration is a physical quantity that has direction. 

In addition to magnitude and direction, a vector quantity must follow rules of vector algebra like vector addition, subtraction, and multiplication. 

Acceleration is a vector as it is always associated with force. Even if we leave force aside if we look at the formula of acceleration we have 

$$\vec a=\frac{\vec v}{dt}$$

where $\vec v$ is the velocity of the moving object and we all know that velocity is defined as displacement of an object w.r.t. time. Mathematically,

$$\vec v=\frac{\vec r}{dt}$$

Here both displacement ($\vec r$) and velocity ($\vec v$) are vector quantities and hence follow laws of vector algebra. 

So, we can conclude that acceleration also follows the laws of vector algebra and is a vector quantity.