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Maths Question 15 – JEE-MAIN 2026

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Let a line L passing through the point (1,1,1) be perpendicular to both the vectors 2i^+2j^+k^ and i^+2j^+2k^. If P(a, b, c) is the foot of perpendicular from the origin on the line L, then the value of 34(a+b+c) is :

If a line is perpendicular to two vectors, its direction vector is parallel to the cross product of those two vectors.

Step 1: Determine the Direction Vector of Line L✦ Active

The line L passes through A(1,1,1) and is perpendicular to v1→=2i^+2j^+k^ and v2→=i^+2j^+2k^. Therefore, its direction vector d→ is parallel to the cross product of v1→ and v2→.

d→=v1→×v2→=|i^j^k^221122|=i^(4−2)−j^(4−1)+k^(4−2)=2i^−3j^+2k^

The equation of line L is r→=(i^+j^+k^)+t(2i^−3j^+2k^). Any point P on the line L can be written as P(1+2t,1−3t,1+2t).

Step 2: Find the Foot of the Perpendicular P(a, b, c)○ Expand

Let P(a, b, c) be the foot of the perpendicular from the origin O(0,0,0) to line L. The vector OP→ must be perpendicular to the direction vector d→ of line L. So, OP→⋅d→=0.

OP→=(1+2t)i^+(1−3t)j^+(1+2t)k^

Applying the dot product condition:

(1+2t)(2)+(1−3t)(−3)+(1+2t)(2)=02+4t−3+9t+2+4t=01+17t=0t=−117

Substitute t=−117 into the coordinates of P:

a=1+2(−117)=1−217=1517b=1−3(−117)=1+317=2017c=1+2(−117)=1−217=1517
💡 Teacher's Secret Hint

Ensure careful calculation of the cross product and the dot product to avoid sign errors.

Step 3: Calculate the Final Value○ Expand

Now, calculate a+b+c:

a+b+c=1517+2017+1517=15+20+1517=5017

Finally, calculate 34(a+b+c):

34(a+b+c)=34×5017=2×50=100
💡 Teacher's Secret Hint

Double-check the arithmetic, especially when dealing with fractions.

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