Question

The magnitude of a vector on the addition of two vectors $6\hat i + 7\hat j$ and $3\hat {i} + 4\hat j$ is
A. $\sqrt {132} $
B. $\sqrt {136} $
C. $\sqrt {160} $
D. $\sqrt {202} $

Answer 1

Hint: In this question, we first take the given two vectors, A and B, and add them, then use the magnitude formula of a vector to find the required result.

Formula Used:
We have been using the following formula to find magnitude:
$\left| m \right| = \sqrt {{{\left( x \right)}^2} + {{\left( y \right)}^2}} $

Complete step by step solution:
 We are given that $6\hat i + 7\hat j$ and $3\hat {i} + 4\hat j$ are two vectors.
Now let us assume that
 $A = 6\hat {i} + 7\hat j$
$B = 3\hat {i} + 4\hat j$
Now we take the addition of two vectors as:
$
  A + B = \left( {6 + 3} \right)\hat i + \left( {7 + 4} \right)\hat j \\
   = 9\hat i + 11\hat j \\
 $
Now we find the magnitude of a vector by using the formula $\left| m \right| = \sqrt {{{\left( x \right)}^2} + {{\left( y \right)}^2}} $:
$
  \left| {A + B} \right| = \sqrt {{{\left( 9 \right)}^2} + {{\left( {11} \right)}^2}} \\
   = \sqrt {81 + 121} \\
   = \sqrt {202} \\
 $
Therefore, the magnitude of a vector on the two additions of vectors is $\sqrt {202} $.

Option ‘D’ is correct

Additional information: As we know, a vector can be defined as an object with both magnitudes and a direction. Now suppose we need to find the magnitude of a vector formula and the length of any given vector. Velocity, displacement, force, momentum, and other vector quantities are examples of vectors. On the other hand, scalar quantities include speed, mass, distance, volume, temperature, and so on. Scalar quantities have only one magnitude, whereas vector quantities typically have both magnitude and direction.

Note: The magnitude of a vector represents its length. When we add and subtract two vectors, we get two different vectors. A vector's magnitude can never be negative. This is due to the modulus symbol converting all negatives to positives. As a result, the magnitude of a vector is always positive.