Part 12 : Vectors

A vectors are one dimensional tensors, we have seen them in Part4B.

Some examples of vectors are

a is a column vector and b is a row vector

Vector could be represented with a bold letter or a right arrow on top of letter.

Vector convey much more information than just magnitude of any quantity, because a vector has magnitude as well as direction. That’s why vectors show up in physics, electrical engineering, data science and many other fields.

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The speed is a scalar quantity (0 dimensional tensor) but velocity is a vector. So, speed can only convey the magnitude of rate of change of distance per unit time but velocity can convey magnitude as well as direction of rate of change of distance per unit time.

Magnitude of Vector

To calculate the magnitude of any vector we just square all the components (elements), add them together and take square root of the sum.

Magnitude of vectors a and b mentioned above

Magnitude of a vector is represented with vector surrounded by modulus operator.

It we multiply the transpose of the vector with itself and then take the square root of that product, we can get the magnitude of vector.

On calculation we get

Taking square root

We yield the same result as we did above.

Unit Vector

A unit vector has magnitude equal to 1.

The unit vectors in three directions (x, y and z) of Cartesian co-ordinate system are denoted as i, j and k with a hat (Circumflex).

Any vector in R³ vector space could be represented using these three unit vector.

Some vectors in R³ vector space

c and d represented on graph

Representation of vectors using unit vectors

Additional Resources

Khan academy has an amazing explanation of vectors here.

Read Part 13 : Vector Components

You can view the complete series here
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