Answer

Verified

387k+ views

**Hint:**We try to find the direction ratios of the vector $\overrightarrow{AB}$ from the given points $A\equiv \left( 3,1,-2 \right);B\equiv \left( -1,0,1 \right)$. We also form the vector in the form of projections on the axes and particular planes. We equate those values and find the solution for $3{{l}^{2}}-m+1$.

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

The given points are $A\equiv \left( 3,1,-2 \right);B\equiv \left( -1,0,1 \right)$.

We try to find the direction ratios of the vector $\overrightarrow{AB}$.

So, the direction ratios will be the differences of the individual coordinates.

Therefore, $\overrightarrow{AB}=\left( 3+1,1-0,-2-1 \right)=\left( 4,1,-3 \right)$. If we express it in the form of vector, we get

$\overrightarrow{AB}=4\widehat{i}+\widehat{j}-3\widehat{k}$.

If we want to find the projection of a vector $\overrightarrow{X}={{a}_{x}}\widehat{i}+{{a}_{y}}\widehat{j}+{{a}_{z}}\widehat{k}$ on individual axes then we take the coefficients of the directions as the projection value. Here the projections on the X, Y and Z axes are ${{a}_{x}},{{a}_{y}},{{a}_{z}}$ respectively.

For projection on a plane, we take the root of the sum of the square of individual projection.

For projection on the Z-X plane, we have $\sqrt{{{a}_{x}}^{2}+{{a}_{z}}^{2}}$.

For the vector $\overrightarrow{AB}=4\widehat{i}+\widehat{j}-3\widehat{k}$, it’s given that l, m are the projections of AB on the Y-axis, Z-X plane respectively.

This means $l={{a}_{y}}=1,m=\sqrt{{{a}_{x}}^{2}+{{a}_{z}}^{2}}=\sqrt{{{4}^{2}}+{{\left( -3 \right)}^{2}}}=5$.

**Now we need to find the value of $ 3{{l}^{2}} - m+1$. We place the values and get**

$3{{l}^{2}}-m+1=3-5+1=-1$. The correct option is A.

$3{{l}^{2}}-m+1=3-5+1=-1$. The correct option is A.

**Note:**We need to remember that the vector projection is the vector produced when one vector is resolved into two component vectors, one that is parallel to the 2nd vector and one that is perpendicular to the 2nd vector. The parallel vector is the vector projection. So, the coefficients remain the same for the parallel vectors and decide the projection value.

Recently Updated Pages

How do you find slope point slope slope intercept standard class 12 maths CBSE

How do you find B1 We know that B2B+2I3 class 12 maths CBSE

How do you integrate int dfracxsqrt x2 + 9 dx class 12 maths CBSE

How do you integrate int left dfracx2 1x + 1 right class 12 maths CBSE

How do you find the critical points of yx2sin x on class 12 maths CBSE

How do you find the general solution to dfracdydx class 12 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

The Buddhist universities of Nalanda and Vikramshila class 7 social science CBSE

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

Give 10 examples for herbs , shrubs , climbers , creepers

Name 10 Living and Non living things class 9 biology CBSE

Which are the Top 10 Largest Countries of the World?

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Who founded the Nalanda University 1 Mauryan 2 Guptas class 6 social science CBSE