JEE Main Vectors & 3D Geometry – Topic Test

Topics in this test

Difficulty: 15 medium · 15 hard

Vectors & 3D Geometry MCQ questions with answers

The first 3 of 30 questions. Unlock the series for the rest, each with a solution.

1. Q1. A line makes angles of 90°, 60° and 30° with the positive directions of the x-, y- and z-axes respectively. The direction cosines of this line are:

   एक रेखा x-, y- और z-अक्षों की धनात्मक दिशाओं के साथ क्रमशः 90°, 60° और 30° के कोण बनाती है। इस रेखा की दिक्-कोज्याएँ हैं:

   • A. 1, 1/2, √3/2
   • B. 0, √3/2, 1/2
   • C. 0, 1/2, 1/3
   • D. 0, 1/2, √3/2
   Solution

   The direction cosines are the cosines of the angles with the axes: cos90° = 0, cos60° = 1/2, cos30° = √3/2, and these satisfy 0 + 1/4 + 3/4 = 1.

   दिक्-कोज्याएँ अक्षों के साथ बने कोणों की कोज्याएँ होती हैं: cos90° = 0, cos60° = 1/2, cos30° = √3/2, और ये 0 + 1/4 + 3/4 = 1 को संतुष्ट करती हैं।

2. Q2. A vector of magnitude 9 in the direction of the vector 2i - j + 2k is:

   सदिश 2i - j + 2k की दिशा में परिमाण 9 वाला सदिश है:

   • A. 18i - 9j + 18k
   • B. 6i - 3j + 6k
   • C. 2i - j + 2k
   • D. (1/3)(2i - j + 2k)
   Solution

   Since |2i − j + 2k| = 3, the unit vector in that direction is (1/3)(2i − j + 2k), and multiplying by 9 gives 6i − 3j + 6k.

   चूँकि |2i − j + 2k| = 3, उस दिशा में इकाई सदिश (1/3)(2i − j + 2k) है, और 9 से गुणा करने पर 6i − 3j + 6k प्राप्त होता है।

3. Q3. The angle between the vectors a = i - 2j + 3k and b = 3i - 2j + k is:

   सदिश a = i - 2j + 3k तथा b = 3i - 2j + k के बीच का कोण है:

   • A. cos⁻¹(5/7)
   • B. cos⁻¹(5/14)
   • C. cos⁻¹(3/7)
   • D. cos⁻¹(10/√14)
   Solution

   cos θ = (a·b)/(|a||b|) = 10/(√14·√14) = 10/14 = 5/7, so θ = cos⁻¹(5/7).

   cos θ = (a·b)/(|a||b|) = 10/(√14·√14) = 10/14 = 5/7, अतः θ = cos⁻¹(5/7)।

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