Vectors, Co – Planarity of Vectors, Resolution of a Vector Along Two Per Perpendicular Axes, Resolution of a Vector in Three Dimensions Along Three Mutually Perpendicular Axes

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Co – Planarity of Vectors:

Given any two non-collinear vectors and , they can be made to lie in one plane. There (in the plane), the vectors will be intersecting.

Co – planarity of Vectors

Co – Planarity of Vectors

Co – planarity of Vectors

We take their common point as and let the two vectors be and . Given a third vector , coplanar with and , we can choose its initial point also as .

Let be its terminal point. With as diagonal complete the parallelogram with and as adjacent sides.

Thus, any , coplanar with and , is expressible as a linear combination of and . i.e.

Resolution of a Vector Along Two Per Perpendicular Axes:

Consider two mutually perpendicular unit vectors and along two mutually perpendicular axes and .

We have seen above that any vector in the plane of and , can be written in the form

Vector along Two per Perpendicular Axes

Vector Along Two Per Perpendicular Axes

Vector along Two per Perpendicular Axes

If is the initial point of , then and and and are called the component vectors of along x-axis and y-axis.

and , in this special case, are also called the resolved parts of .

Resolution of a Vector in Three Dimensions Along Three Mutually Perpendicular Axes:

The concept of resolution of a vector in three dimensions along three mutually perpendicular axes is an extension of the resolution of a vector in a plane along two mutually perpendicular axes.

Any vector in space can be expressed as a linear combination of three mutually perpendicular unit vectors and as is shown in the adjoining Fig.

Three Dimensions along Three Mutually Perpendicular Axes

Three Dimensions Along Three Mutually Perpendicular Axes

Three Dimensions along Three Mutually Perpendicular Axes

We complete the rectangular parallelepiped with as its diagonal then

and are called the resolved parts of along three mutually perpendicular axes.

Thus any vector in space is expressible as a linear combination of three mutually perpendicular unit vectors and .

In which (Two dimensions)

Or ….. (1)

In Fig.

Or ….. (2)

Magnitude of in case of (1) is

And (2) is

Example:

Show that the following vectors are coplanar: , and

Solution:

The vectors will be coplanar if there exists scalars and such that

….. (1)

Comparing the co-efficient of and on both sides of (1), we get

and

Which on solving, gives and

As a is expressible in terms of and , hence the three vectors are coplanar.

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