Given two vectors are $ \hat{i}-\hat{j} $ and $ \hat{i}+2\hat{j}, $ the unit vector is coplanar with the two vectors and perpendicular to the first. Find the vector?
- $ +\frac{1}{\sqrt{2}}\,(i+k) $
- $ +\frac{1}{\sqrt{5}}\,(2i+j) $
- $ +\frac{1}{\sqrt{2}}\,(i+j) $
- $ +\frac{1}{\sqrt{2}}\,(2i+3j) $
The Correct Option is C
Solution and Explanation
Let given two vectors are \(a=i-j\) and \(b=i+2j\)
Again let third unit vector is c.
\(\because\) c is coplanar with a, b
\(\therefore\) \(c=xa+yb\)
\(=x(i-j)+y(i+2j)\)
\(\Rightarrow\) \(c=(a+y)i+(-x+2y)j\) ..(i)
Also, c is perpendicular to a \(\therefore\)
\(a.c=0\)
\(\Rightarrow\) \((i-j).\{(x+y)i+(-x+2y)j\}=0\)
\(\Rightarrow\) \((x+y)-(-x+2y)=0\)
\(\Rightarrow\) \(x+y+x-2y=0\)
\(\Rightarrow\) \(2x-y=0\Rightarrow y=2x\)
On putting this value of y in E (i), we get
\(c=(x+2x)i+(-x+4x)j\)
\(\Rightarrow\) \(c=3xi+3xj\) ..(ii)
But c is a unit vector. So, \(|c|=1\)
\(\Rightarrow\) \({{(3x)}^{2}}+{{(3x)}^{2}}=1\,\Rightarrow 9{{x}^{2}}+9{{x}^{2}}=1\)
\(\Rightarrow\) \({{x}^{2}}=\frac{1}{18}\,\,\,\Rightarrow \,\,\,x=\frac{1}{3\sqrt{2}}\)
On putting this value of x in E (ii), we get \(c=3.\frac{1}{3\sqrt{2}}i+3.\frac{1}{3\sqrt{2}}j\,\,\Rightarrow c=\frac{1}{\sqrt{2}}\,(i+j)\)
Coplanar vectors are those in a three-dimensional plane that are parallel to one another. In the same plane, the vectors are perpendicular to one another. Any two random vectors on a plane may always be in line with one another. In a three-dimensional space that can be described in vector form, two lines are coplanar. When the scalar product of three vectors is equal to zero, the vectors are referred to as being planar.
The coplanar line is a well-known concept in three-dimensional geometry. In mathematical theory, three lines that are situated on the same plane are referred to as being coplanar and this property is known as the coplanarity of three vectors. Although straight lines have been used as vector equations, a plane is a two-dimensional geometry that stretches into infinity in three-dimensional space.
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Concepts Used:
Vector Algebra
A vector is an object which has both magnitudes and direction. It is usually represented by an arrow which shows the direction(→) and its length shows the magnitude. The arrow which indicates the vector has an arrowhead and its opposite end is the tail. It is denoted as
The magnitude of the vector is represented as |V|. Two vectors are said to be equal if they have equal magnitudes and equal direction.
Vector Algebra Operations:
Arithmetic operations such as addition, subtraction, multiplication on vectors. However, in the case of multiplication, vectors have two terminologies, such as dot product and cross product.