Question

Write all the unit vectors in the XY plane.

Answer 1

Hint: A vector is a quantity that has both direction and magnitude as well. A vector having magnitude of 1 is known as the unit vector. As its magnitude is 1, they are also known as the direction vectors. A unit vector is denoted as \[\mathop a\limits^ \to = x\mathop i\limits^ \wedge + y\mathop j\limits^ \wedge + z\mathop k\limits^ \wedge \]. Here, as we have to find the unit vectors in XY plane, the z coordinate will be 0.

Complete step-by-step answer:
In this question, we are asked to write all the unit vectors in the XY plane.
First of all, let us see what vectors are.
A vector is a quantity that has both direction and magnitude as well. A vector having magnitude of 1 is known as the unit vector. As its magnitude is 1, they are also known as the direction vectors.
A unit vector is represented by ‘^’, which is called a cap or hat.
For example: $\mathop x\limits^ \wedge $
$ \Rightarrow \mathop x\limits^ \wedge = \dfrac{x}{{\left| x \right|}}$
Where, $\left| x \right| = $magnitude of x.
Now, let our unit vector be
\[\mathop a\limits^ \to = x\mathop i\limits^ \wedge + y\mathop j\limits^ \wedge + z\mathop k\limits^ \wedge \], where \[\mathop i\limits^ \wedge \], \[\mathop j\limits^ \wedge \] and \[\mathop k\limits^ \wedge \] are the direction vectors along X – Axis , Y – Axis and Z – Axis respectively.
The magnitude of this is given by
\[ \Rightarrow \mathop {\left| a \right|}\limits^ \to = \dfrac{{x\mathop i\limits^ \wedge + y\mathop j\limits^ \wedge + z\mathop k\limits^ \wedge }}{{\sqrt {{x^2} + {y^2} + {z^2}} }}\]
Here, we have to find the unit vectors in the XY plane.
XY Plane means 1st quadrant. In the 1st quadrant z coordinate is 0. Therefore, we get
\[ \Rightarrow \mathop a\limits^ \to = x\mathop i\limits^ \wedge + y\mathop j\limits^ \wedge \] and
\[ \Rightarrow \mathop {\left| a \right|}\limits^ \to = \dfrac{{x\mathop i\limits^ \wedge + y\mathop j\limits^ \wedge }}{{\sqrt {{x^2} + {y^2}} }}\]
Hence, all the unit vectors in XY plane are given by \[\mathop {\left| a \right|}\limits^ \to = \dfrac{{x\mathop i\limits^ \wedge + y\mathop j\limits^ \wedge }}{{\sqrt {{x^2} + {y^2}} }}\].

Note: We can also find the unit vectors for other planes.
In XZ Plane the y coordinate will be 0. So,
\[ \Rightarrow \mathop a\limits^ \to = x\mathop i\limits^ \wedge + z\mathop k\limits^ \wedge \] and
\[ \Rightarrow \mathop {\left| a \right|}\limits^ \to = \dfrac{{x\mathop i\limits^ \wedge + z\mathop k\limits^ \wedge }}{{\sqrt {{x^2} + {z^2}} }}\]
In the YZ plane the x coordinate will be 0. So,
\[ \Rightarrow \mathop a\limits^ \to = y\mathop j\limits^ \wedge + z\mathop k\limits^ \wedge \] and
\[ \Rightarrow \mathop {\left| a \right|}\limits^ \to = \dfrac{{y\mathop j\limits^ \wedge + z\mathop k\limits^ \wedge }}{{\sqrt {{y^2} + {z^2}} }}\]