Now that you’ve understood what vectors are, let’s go ahead and learn about some vector properties and a few associated operations:

Let’s summarise the learnings of the above lecture.

**Vectors have a direction and magnitude**

Each vector has a direction and magnitude associated with it. The direction is given by an arrow starting from the origin and pointing towards the vector’s position. The magnitude is given by taking a sum of squares of all the coordinates of that vector and then taking its square root.

For example, the vector (2,3) has the direction given by the arrow joining (0,0) and (2,3) pointing towards (2,3). Its magnitude is given by √22+32=√13.

Similarly, for a vector in 3 dimensions, say (2,-3,4) its direction is given by the arrow joining (0,0,0) and (2,-3,4) pointing towards (2,-3,4). And as in the 2D case, we get the magnitude of this vector as √(2)2+(−3)2+(4)2=√29 .

**Vector Addition**

When you add two or more vectors, we essentially add their corresponding values element-wise. The first element of both the vectors get added, the second element of the both get added and so on.

For example, if you’ve two vectors say

V1=(2,3) and V2=(1,2) then

V1+V2=(2+1,3+2)=(3,5).

- In the
**i, j**notations that we introduced earlier, the above addition can be written as V1+V2=(2i+3j)+(i+2j)=(2+1)i+(3+2)j=3i+5j

Similarly, this idea can be extended to multiple dimensions as well.

**Scalar Multiplication**

If you multiply any real number or scalar by a vector, then there is a change in the magnitude of the vector and the direction remains same or turns completely opposite depending on whether the value is positive or negative respectively.

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