ভেক্টরের স্কেলার গুণন

Given a vector $\bar{v}$ with coordinates $(x,y)$, multiply it by scalar $k$. Now what's a scalar? A $scalar$ is just a constant, so we are scaling this vector by a constant $k$. Now how do we do that? Just multiply the coordinates of $\bar{v}$ by $k$.

Now practice your new skill by multiplying these vectors below:

We can multiply a vector by a scalar in exactly the same way as multiplying two scalars: repeated addition!

For example, to multiply a vector $\vec{a}$ by a scalar $k$: $$k\vec{a}=\underbrace{\vec{a}+\vec{a}+...+\vec{a}}_{\text{k terms}}$$ This is all nice, but what does it mean?

Let us say that we know the components of the vector. That is, suppose that we have a vector in $n$-dimensional coordinate space: $$\vec{a}=\left<a_1,a_2,...,a_n\right>$$ Multiplying this vector by a scalar $k$, $$k\vec{a}=k\left<a_1,a_2,...,a_n\right>$$ $$k\vec{a}=k\left<a_1,a_2,...,a_n\right>$$ $$k\vec{a}=\underbrace{\left<a_1,a_2,...,a_n\right>+\left<a_1,a_2,...,a_n\right>+...+\left<a_1,a_2,...,a_n\right>}_{\text{k terms}}$$ $$k\vec{a}=\left<ka_1,ka_2,...,ka_n\right>$$ This means that, to multiply a vector by a scalar, you should multiply all of its components by that scalar.

Also, since the $k$ copies of the vector $\vec{a}$ all point in the same direction, the vector $k\vec{a}$ is parallel to $\vec{a}$. In general, multiplying a vector by a scalar gives us a vector parallel to the original.

In fact, the converse is also true: if a vector is parallel to another, then it is a scalar multiple of the other.

Note that if $k>0$, then the two vectors point in the same direction, while if $k<0$, then the two vectors point in opposite directions.

If $k=0$, then one vector is the zero vector, which by this convention is considered parallel to all vectors. This is strange, but it is usually accepted as a quirk of the system.

Also, by using components, it is easy to prove that the vector $k\vec{a}$ is $|k|$ times as long as $\vec{a}$: $$|k\vec{a}|=\sqrt{(ka_1)^2+(ka_2)^2+...+(ka_n)^2}$$ $$|k\vec{a}|=\sqrt{k^2(a_1^2+a_2^2+...+a_n^2)}$$ $$|k\vec{a}|=|k|\sqrt{a_1^2+a_2^2+...+a_n^2}$$ $$|k\vec{a}|=|k||\vec{a}|$$ In summary, multiplying a vector by a scalar produces a vector parallel to the original, pointing in a direction determined by the sign of the scalar. The length of the new vector is equal to the length of the original times the absolute value of the scalar. If the vector is expressed in components, then the new vector is made by multiplying every component by the scalar.