Vectors and Their General Properties

Vectors and their general properties

We already learned about the vectors and their notations in the previous article. Now we will learn about the general properties of vectors, and we will see what a unit vector is.

Equality of Vectors:

When we are talking about the equality of two vectors, they must represent the same physical quantity. Two vectors are said to be equal if their magnitudes and directions are the same. Therefore, we can conclude that a parallel translation of a vector does not bring about any change in it.

Multiplication of a vector by a Number:

Suppose \( \overrightarrow {a}\)   is a vector of magnitude ‘a’ and n is a number.  If we multiply \( \overrightarrow {a}\) by n, then we will get a new vector. Let’s say this vector is  \( \overrightarrow {b}\)

We define vector \( \overrightarrow {b}\) = n \( \overrightarrow {a}\) as the vector of magnitude na. The direction of vector  \( \overrightarrow {b}\) will be the same as of vector \( \overrightarrow {a}\)  i.e. if the vector \( \overrightarrow {a}\) is in positive x direction, vector \( \overrightarrow {b}\) will be in positive x direction; if the vector \( \overrightarrow {a}\)  is in negative x direction, vector \( \overrightarrow {b}\)  will be in negative x direction.

If this number n is negative, the direction of \( \overrightarrow {b}\) is opposite to the direction of vector \( \overrightarrow {a}\). If n = -1, then multiplication by n just reverses the direction of the vector.

Unit vector:

A unit vector is a vector of magnitude (or length) 1 unit. Thus, unit vectors are used to specify the directions of vector quantities in various coordinate systems. In Cartesian coordinates, generally:

\( \hat {i} \)  = unit vector in x-direction

\( \hat {j} \)  = unit vector in y-direction

\( \hat {k} \)  = unit vector in z direction

Position vector of any object can be represented in Cartesian coordinates as:

\( \overrightarrow {r}\) = \( x \hat{i}~+~ y \hat{j}~+~ x \hat{k} \)

There is a lot more to know about vectors. Knowing about vectors is the first and basic step towards learning higher physics as it deals with a lot of vector quantities. Stay tuned with BYJU’S to learn more about vectors, vectors notations much more.

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