NIMCET 2010 Mathematics PYQ — The vectors and are equal in length and taken pairwise make equal… | Mathem Solvex | Mathem Solvex
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NIMCET 2010 — Mathematics PYQ
NIMCET | Mathematics | 2010
The vectors a,b and c are equal in length and taken pairwise make equal angles. If a=i^+j^,b=j^+k^, and c make an obtuse angle with the base vector i^, then c is equal to:
Choose the correct answer:
A.
i^+k^
B.
−i^+4j^−k^
C.
−31i^+34j^−31k^
D.
31i^+34j^−31k^
Correct Answer:
−31i^+34j^−31k^
Explanation
Step 1: Determine the magnitude and common angle.
Given a=i^+j^ and b=j^+k^.
Magnitude ∣a∣=12+12=2.
Magnitude ∣b∣=02+12+12=2.
Since all vectors have equal length, let c=xi^+yj^+zk^. Then ∣c∣2=x2+y2+z2=2.
The dot product a⋅b=(1)(0)+(1)(1)+(0)(1)=1.
Since they make equal angles pairwise, the dot products must be equal:
a⋅b=b⋅c=c⋅a=1
Step 2: Set up equations for c.
Using the dot product conditions:
a⋅c=(1)x+(1)y+(0)z=1⟹x+y=1…(Eq. 1)
b⋅c=(0)x+(1)y+(1)z=1⟹y+z=1…(Eq. 2)
From these, we can express x and z in terms of y:
x=1−y
z=1−y
Step 3: Use the magnitude condition.
Substitute x and z into x2+y2+z2=2:
(1−y)2+y2+(1−y)2=2
(1−2y+y2)+y2+(1−2y+y2)=2
3y2−4y+2=2
3y2−4y=0⟹y(3y−4)=0
So, y=0 or y=34.
Step 4: Check the "obtuse angle" condition.
The vector c makes an obtuse angle with i^ if \vec{c} \cdot \hat{i} < 0.
c⋅i^=x. Therefore, we need x < 0.
Case 1: y=0
x=1−0=1. (Not obtuse, as 1 > 0)
Case 2: y=34
x=1−34=−31. (Obtuse, as -\frac{1}{3} < 0)
z=1−34=−31.
Step 5: Final Vector.
Substituting x,y,z values:
c=−31i^+34j^−31k^
Correct Option: (c) −31i^+34j^−31k^
Explanation
Step 1: Determine the magnitude and common angle.
Given a=i^+j^ and b=j^+k^.
Magnitude ∣a∣=12+12=2.
Magnitude ∣b∣=02+12+12=2.
Since all vectors have equal length, let c=xi^+yj^+zk^. Then ∣c∣2=x2+y2+z2=2.
The dot product a⋅b=(1)(0)+(1)(1)+(0)(1)=1.
Since they make equal angles pairwise, the dot products must be equal:
a⋅b=b⋅c=c⋅a=1
Step 2: Set up equations for c.
Using the dot product conditions:
a⋅c=(1)x+(1)y+(0)z=1⟹x+y=1…(Eq. 1)
b⋅c=(0)x+(1)y+(1)z=1⟹y+z=1…(Eq. 2)
From these, we can express x and z in terms of y:
x=1−y
z=1−y
Step 3: Use the magnitude condition.
Substitute x and z into x2+y2+z2=2:
(1−y)2+y2+(1−y)2=2
(1−2y+y2)+y2+(1−2y+y2)=2
3y2−4y+2=2
3y2−4y=0⟹y(3y−4)=0
So, y=0 or y=34.
Step 4: Check the "obtuse angle" condition.
The vector c makes an obtuse angle with i^ if \vec{c} \cdot \hat{i} < 0.
