Question

Resultant of two vectors $\vec A = 3\hat i + 4\hat j$ and $\vec B = \alpha \hat i + 3\hat j$ lies along the Y – axis. The value of $\alpha $ is
A) 3
B) -3
C) 4
D) -4

Answer 1

Hint: The resultant of the given two vectors is given by their sum. If the resultant lies along y – axis, its component along x – axis will be zero. Using this information, we can find the unknown value of $\alpha $.

Complete Step by step answer:The given vectors are:
$\vec A = 3\hat i + 4\hat j$ and
$\vec B = \alpha \hat i + 3\hat j$
Now, the resultant $\left( {\vec R} \right)$ of these vectors is given by their sum:
$\vec R = \vec A + \vec B$

Substituting the values, we get:
$
  \vec R = \left( {3\hat i + 4\hat j} \right) + \left( {\alpha \hat i + 3\hat j} \right) \\
 \Rightarrow \vec R = \left( {3 + \alpha } \right)\hat i + \left( {4 + 3} \right)\hat j \\
 \Rightarrow \vec R = \left( {3 + \alpha } \right)\hat i + 7\hat j \\
 $
Where $\hat i$ represents x – axis and $\hat j$ represents y – axis.
According to the question, the resultant of these vectors lies along y- axis, which means the component along x - axis will be equal to zero.
$
   \Rightarrow 3 + \alpha = 0 \\
  \alpha = - 3 \\
 $ [component along x – axis]
Therefore, the unknown value of $\alpha $ in the vector B is -3 and the correct option is B.

Additional information: Vectors are inclined at certain angles which give their direction. The starting point is called tail and ending as head.
Velocity and force are most commonly used vector quantities.
The vector with magnitude 1 is called unit vector and 0 magnitude is called zero vector.
The magnitude of the vector can be calculated either by the Pythagoras theorem or directly taking the square root of the sum of squares of the components along the respective axes.

Note: If two vectors have the same direction, the resultant of these vectors will be in the opposite direction when represented with the help of line segments.
$\hat i$ and $\hat j$ are called unit vectors (magnitude 1) and are used to represent the direction of vectors along the x and y- axis respectively.
The quantities that have both direction and the magnitude are called vectors and they are represented by an arrow over the head.