Answer

Verified

447.6k+ views

**Hint:**The projection of a point is its shadow on the plane or central projection.

If C is a point, called the centre of projection then the projection of a point P different from C onto a plane that does not contain C is the interaction of the line CP with the plane.

**Complete step-by- step solution:**

Let us draw a plane and the projection of point \[P(\mathop p\limits^ \to )\]on the plane is \[\mathop s\limits^ \to \].

The intersection is \[\mathop r\limits^ \to .\mathop n\limits^ \to = q\]

As the line is normal to the plane i.e. perpendicular to the plane and vector \[\mathop P\limits^ \to \] is passing through the plane and parallel to \[\mathop n\limits^ \to \]

\[E{q^n}\] of such a line is\[\overrightarrow r = \overrightarrow p + \lambda \overrightarrow n ……...(1)\]

Given, \[\mathop r\limits^ \to .\mathop n\limits^ \to = q........(2)\]

As the line is passing through the plane, then the equation (1) will be satisfying equation (2) and that point \[\mathop r\limits^ \to = \mathop s\limits^ \to \]

Substituting equation (1) in (2), we get:

$\Rightarrow$ \[(\mathop p\limits^ \to + \lambda \mathop n\limits^ \to )\mathop n\limits^ \to = q\]

To find the value of \[\lambda \], simplify the above term then we get it as

\[ \Rightarrow \mathop p\limits^ \to . \mathop n\limits^ \to + \lambda \mathop n\limits^ \to .\mathop n\limits^ \to = q\]

As \[\left[ {\overrightarrow n .\overrightarrow n = {{\left| {\mathop n\limits^ \to } \right|}^2}} \right]\] , we get:

\[ \Rightarrow \lambda {\left| {\mathop n\limits^ \to } \right|^2} = q - \mathop p\limits^ \to .\mathop n\limits^ \to \]

\[ \Rightarrow \lambda = \dfrac{{q - \mathop p\limits^ \to .\mathop n\limits^ \to }}{{{{\left| {\mathop n\limits^ \to } \right|}^2}}}\]______ (3) {On RHS \[{\left| {\mathop n\limits^ \to } \right|^2}\] will be in division as it was multiplication on LHS}

Now using equation (3) in (1), we get:

\[\mathop r\limits^ \to = \mathop p\limits^ \to + (\dfrac{{q - \mathop p\limits^ \to .\mathop n\limits^ \to }}{{{{\left| {\mathop n\limits^ \to } \right|}^2}}})\mathop n\limits^ \to \]

We know that\[\overrightarrow r = \overrightarrow s \], hence:

\[\mathop s\limits^ \to = \mathop p\limits^ \to + (\dfrac{{q - \mathop p\limits^ \to .\mathop n\limits^ \to }}{{{{\left| {\mathop n\limits^ \to } \right|}^2}}})\mathop n\limits^ \to \]

**Note:**Two planes are parallel if they have the same normal vector (i.e. their normal vectors are parallel). If two planes are not parallel, then they intersect in a line.If any line passes through a plane then it always satisfies the equation of that plane.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Which are the Top 10 Largest Countries of the World?

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Change the following sentences into negative and interrogative class 10 english CBSE