Why is acceleration a vector quantity?

Hint: In order to answer this question, we will be studying what we mean by the vectors and scalars. After getting some basic idea we will be taking acceleration into consideration and finally reach our answer on why acceleration is a vector quantity.

Complete step by step solution:
First of all, we will understand the terms vector and scalar.
In physics, a vector is a quantity with both magnitude and direction. It's usually represented by an arrow with the same direction as the quantity and a length equal to the magnitude of the quantity. A vector does not have position, even though it has magnitude and direction. A vector is not affected if it is displaced parallel to its origin as long as its length is not changed.
Ordinary quantities with a magnitude but no direction are called scalars, in contrast to vectors. Displacement, velocity, and acceleration, for example, are vector quantities, while speed (velocity magnitude), time, and mass are scalars.
Since acceleration has both magnitude and direction, acceleration is a vector quantity.
When an object accelerates in the same direction as it moves, it is said to have a positive acceleration. When an object slows down (has a negative acceleration), the acceleration happens in the opposite direction of the object's movement.
Consider a ball being tossed into the air. The ball is being accelerated by gravity at a steady rate of $g=9.8\dfrac{m}{{{s}^{2}}}$. When the ball is moving upward, the acceleration is in the opposite direction, causing the ball to slow down. Gravity continues to work on the ball even though it slows to a velocity of zero. The ball then starts to fall because gravity continues to work on it, but now the motion and acceleration are in the same direction, causing the ball to accelerate.

Note:
It is very important to note that apart from having magnitude and direction, a physical quantity must also satisfy certain rules of vector combination. The most common rules of vector combination are vector addition, vector subtraction, scalar multiplication, vector multiplication and differentiation. A physical quantity must satisfy all the above operations to get classified as a vector quantity.