How to multiply a vector by a scalar
Vectors and matrices are mathematical objects that have many uses in physics and engineering. In this blog post, we'll focus on vectors. In particular, we'll explain what it means to multiply a vector by a scalar.
Vectors can be multiplied by scalars in one of two ways: the dot product or the cross product. The dot product is the most common way to multiply vectors, and it results in a scalar value. To take the dot product of two vectors, you need to take the sum of the products of the corresponding components of each vector. For example, if you have the following two vectors:
Vector A = [1, 2, 3] Vector B = [4, 5, 6] Then the dot product of vector A and vector B would be 1*4 + 2*5 + 3*6 = 32. The other way to multiply vectors is called the cross product. The cross product results in a vector, which is perpendicular to both of the original vectors. To take the cross product of two vectors, you need to take the determinant of the following matrix: $$\begin{bmatrix} i & j & k \\ \vec{A}_1 & \vec{A}_2 & \vec{A}_3 \\ \vec{B}_1 & \vec{B}_2 & \vec{B}_3\end{bmatrix} $$ For example, if you have the following two vectors: Vector A = [1, 2, 3] Vector B = [4, 5, 6] Then the cross product of vector A and vector B would be $$\begin{vmatrix} i & j & k \\ 1& 2 & 3 \\ 4& 5 & 6\end{vmatrix} = i(2*6-3*5) - j(1*6-3*4) + k(1*5-2*4) $$ $$= (2*6-3*5)*i - (1*6-3*4)*j + (1*5-2*4)*k $$ $$= 12i - 9j + 6k $$
In conclusion, there are two ways to multiply vectors: the dot product and the cross product. The dot product is more common and it results in a scalar value. The cross product results in a vector that is perpendicular to both of the original vectors."
FAQ
How a vector is multiplied by scalar?
When a vector is multiplied by a scalar, each component of the vector is multiplied by the scalar. So, if we have a vector $\vec{v} = (v_1, v_2, \ldots, v_n)$ and a scalar $c$, then the product $\vec{v}c$ is the vector $(cv_1, cv_2, \ldots, cv_n)$.
This operation is also sometimes called "scaling" the vector, because it changes the magnitude of the vector. If $c>0$, then the product vector will have the same direction as the original vector, but it will be longer by a factor of $c$. If $c<0$, then the product vector will have the opposite direction of the original vector, and it will be shorter by a factor of $|c|$. If $c=0$, then the product vector is the "zero vector", which has all zero components.
Can we multiply any vector by any scalar?
Yes, we can multiply any vector by any scalar. This operation is well-defined and always gives a valid result.What are some applications of multiplying vectors by scalars?There are many applications of multiplying vectors by scalars. One common application is in physics, when we need to change the direction or magnitude of a physical quantity that is represented by a vector. For example, when an object is moving in a straight line at a constant speed, we can represent its position at any time $t$ by the vector $\vec{r}(t) = (x(t), y(t), z(t))$. If we want to change the speed of the object, we can simply multiply this vector by a scalar factor $c$ to get a new vector $\vec{r}'(t) = \vec{r}(t)c$.
How do you multiply a scalar product of two vectors?
The answer depends on what you mean by "multiply". If you want to multiply the scalar product of two vectors, you can simply multiply the two scalar factors together. So, if we have a scalar product $\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + \ldots + u_nv_n$, then we can multiply it by a scalar $c$ to get $(\vec{u} \cdot \vec{v})c = (u_1v_1 + u_2v_2 + \ldots + u_nv_n)c$.
However, if you want to multiply the two vectors themselves, then you need to use the cross product. The cross product of two vectors $\vec{u}$ and $\vec{v}$ is a vector $\vec{u} \times \vec{v}$ that is perpendicular to both $\vec{u}$ and $\vec{v}$, and its magnitude is equal to the product of the magnitudes of $\vec{u}$ and $\vec{v}$ divided by the product of their lengths. So, if we have two vectors $\vec{u} = (u_1, u_2, \ldots, u_n)$ and $\vec{v} = (v_1, v_2, \ldots, v_n)$, then the cross product is $\vec{u} \times \vec{v} = (u_2v_3 - u_3v_2,\; u_3v_1 - u_1v_3,\; \ldots,\; u_1v_2 - u_2v_1)$.
Note that the cross product is only defined for three-dimensional vectors. For two-dimensional vectors, we can simply take the cross product to be zero.
