Question

Find the values of x and y so that the vectors \[2\overset{\wedge }{\mathop{i}}\,+3\overset{\wedge }{\mathop{j}}\,\] and \[x\overset{\wedge }{\mathop{i}}\,+y\overset{\wedge }{\mathop{j}}\,\] are equal.

Answer 1

Hint: In this question, we first need to know about the equality of vectors. As we know that equal vectors should have the same length, same or parallel support and the same sense. So, on comparing both of them we get the answer.

Complete step-by-step solution -
Let us look at some of the basic definitions of vectors.
A vector has direction and magnitude both but scalar has only magnitude.
Example: vector quantities are displacement, velocity, acceleration, etc and scalar quantities are length, mass, time, etc.
CHARACTERISTICS OF A VECTOR:
Magnitude: The length of the vector AB or a is called the magnitude of AB or a and it is represented as \[\left| AB \right|or\left| a \right|\].
Sense: The direction of a line segment from its initial point to its terminal point is called its sense.
Example: The sense of AB is from A to B and the sense of BA is from B to A.
Support: The line of infinite length of which the line segment is a part, is called the support of the vector.
Position Vector of a Point: The position vector of a point P with respect to a fixed point, say O, is the vector OP. The fixed point is called origin.
Let PQ be any vector. We have,
\[PQ=PO+OQ=-OP+OQ=OQ-OP\]
PQ = Position vector of Q - Position vector of P.
COLLINEAR POINTS: Let A, B, C be any three collinear points.
Points A, B, C are collinear \[\Leftrightarrow \] AB, BC are collinear vectors.
\[\Leftrightarrow AB=\lambda BC\]for some non-zero scalar \[\lambda \].
Equality of Vectors: Two vectors a and b are said to be equal written as a = b, if they have
Same length
The same or parallel support
The same sense.
Now, given that the two vectors are equal.
\[\begin{align}
  & 2\overset{\wedge }{\mathop{i}}\,+3\overset{\wedge }{\mathop{j}}\, \\
 & x\overset{\wedge }{\mathop{i}}\,+y\overset{\wedge }{\mathop{j}}\, \\
\end{align}\]
\[\Rightarrow 2\overset{\wedge }{\mathop{i}}\,+3\overset{\wedge }{\mathop{j}}\,=x\overset{\wedge }{\mathop{i}}\,+y\overset{\wedge }{\mathop{j}}\,\]
Now, on comparing the \[\overset{\wedge }{\mathop{i}}\,\] component and the\[\overset{\wedge }{\mathop{j}}\,\]component correspondingly,
\[\begin{align}
  & \Rightarrow 2\overset{\wedge }{\mathop{i}}\,=x\overset{\wedge }{\mathop{i}}\, \\
 & \Rightarrow 3\overset{\wedge }{\mathop{j}}\,=y\overset{\wedge }{\mathop{j}}\, \\
 & \therefore x=2,y=3 \\
\end{align}\]

Note: If we equate magnitude first then there will be more than one possibility to find x and y. But, if we consider the sense first and then equate then we get the above values of x and y which in turn has the magnitude also equal.
Vectors are said to be like when they have the same direction and unlike when they have opposite directions. Similarly, vectors having the same or parallel supports are called collinear vectors.