If two vectors P = \(\hat{i} \)+2m\(\hat{j}\)+m\(\hat{k}\) & Q = 4\(\hat{i}\)-2\(\hat{j}\)+m\(\hat{k}\) are perpendicular to each other, then the value of m will be:
Show Hint
For perpendicular vectors, the dot product \( \vec{A} \cdot \vec{B} = 0 \). Expand terms carefully to solve for the unknown.
m = -3
m = 2
m = -1
m = 1
The Correct Option is B
Approach Solution - 1
The condition for two vectors to be perpendicular is:
\[ \vec{P} \cdot \vec{Q} = 0 \]
Substitute the given vectors:
\[ (i + 2m j + m k) \cdot (4i - 2j + m k) = 0 \]
Expanding the dot product:
\[ 1 \cdot 4 + (2m)(-2) + (m)(m) = 0 \]
\[ 4 - 4m + m^2 = 0 \]
Rearranging:
\[ m^2 - 4m + 4 = 0 \]
\[ (m - 2)^2 = 0 \implies m = 2 \]
Thus, the value of \( m \) is 2.
Approach Solution -2
A vector is a physical quantity that has both magnitude and direction and obeys the triangle law or parallelogram law of addition. It is represented by an arrow (→), with the length of the arrow representing the magnitude of the vector and the direction of the arrow representing the direction of the vector.
We have given two vectors
- P = î + 2mĵ + mk̂
- Q = 4î - 2ĵ + mk̂
Taking the dot product of both the vectors, we get
P.Q = PQ cosθ
Where θ is the angle between the vectors P and Q.
Since it is given that both vectors are perpendicular to each other, therefore θ = 900
⇒ P.Q = PQ cos900
⇒ P.Q = 0
On substituting the values, we get
(î + 2mĵ + mk̂ ).(4î - 2ĵ + mk̂) = 0
⇒ (î . 4î ) + (î . - 2ĵ ) + (î . mk̂) + (2mĵ . 4î ) + (2mĵ . - 2ĵ) + (2mĵ mk̂) + (mk̂ . 4î) + (mk̂ . - 2ĵ) + (mk̂ . mk̂) = 0
Now, the dot product of two similar unit vector is 1 and that of the two different unit vectors is 0. Therefore
4 - 4m + m2 = 0
⇒ m2 - 4m + 4 = 0
⇒ m2 - 2m - 2m + 4 = 0
⇒ m(m - 2) - 2(m - 2) = 0
⇒ (m - 2)(m - 2) = 0
⇒ m = 2
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Concepts Used:
Vectors
The quantities having magnitude as well as direction are known as Vectors or Vector quantities. Vectors are the objects which are found in accumulated form in vector spaces accompanying two types of operations. These operations within the vector space include the addition of two vectors and multiplication of the vector with a scalar quantity. These operations can alter the proportions and order of the vector but the result still remains in the vector space. It is often recognized by symbols such as U ,V, and W
Representation of a Vector :
A line having an arrowhead is known as a directed line. A segment of the directed line has both direction and magnitude. This segment of the directed line is known as a vector. It is represented by a or commonly as AB. In this line segment AB, A is the starting point and B is the terminal point of the line.
Types of Vectors:
Here we will be discussing different types of vectors. There are commonly 10 different types of vectors frequently used in maths. The 10 types of vectors are:
- Zero vector
- Unit Vector
- Position Vector
- Co-initial Vector
- Like and Unlike Vectors
- Coplanar Vector
- Collinear Vector
- Equal Vector
- Displacement Vector
- Negative of a Vector