To multiply the vector $(2, -3)$ by the scalar $4$, you need to multiply each component of the vector by $4$. This results in the new vector $(8, -12)$.

When you perform scalar multiplication, you take a vector and multiply each of its components by a single number, known as the scalar. In this instance, the vector is $(2, -3)$ and the scalar is $4$.

First, consider the x-component of the vector, which is $2$. When we multiply it by $4$, we have:

Next, we look at the y-component of the vector, which is $-3$. Multiplying this by $4$ gives us:

Thus, after multiplying the vector $(2, -3)$ by $4$, we obtain the new vector $(8, -12)$. This operation is referred to as scalar multiplication, which scales the vector by the specified factor—in this case, $4$. While the direction of the vector remains unchanged, its magnitude (or length) is increased by a factor of $4$.

Scalar multiplication is a crucial operation in many fields, including physics, where it is often necessary to scale quantities such as forces or velocities. It is also widely used in computer graphics for resizing objects. Mastering scalar multiplication is an essential skill in the study of vector mathematics.