Vector Algebra and 3D Geometry Practice Questions
JEE · Mathematics · 1678 free MCQs with instant results and detailed explanations.
1678
Total
384
Easy
718
Medium
576
Hard
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Topics in Vector Algebra and 3D Geometry
Vectors
473
Scalar and Cross Product
521
Lines and Planes in 3D
684
Sample Questions from Vector Algebra and 3D Geometry
Here are 10 sample questions. Start a quiz to get randomized questions with scoring.
Q1
Easy
What is the magnitude of the vector A = 3i + 4j?
A.
5
B.
7
C.
1
D.
12
Show Answer & Explanation
Correct Answer: A
The magnitude is calculated using the formula โ(xยฒ + yยฒ). Here, โ(3ยฒ + 4ยฒ) = โ(9 + 16) = โ25 = 5.
Q2
Easy
What is the unit vector in the direction of A = 4i - 3j?
A.
(4/5)i - (3/5)j
B.
(3/5)i + (4/5)j
C.
(-4/5)i + (3/5)j
D.
(1/5)i + (1/5)j
Show Answer & Explanation
Correct Answer: A
The unit vector is obtained by dividing the vector by its magnitude. Magnitude = 5, so unit vector = (4/5)i - (3/5)j.
Q3
Easy
If the position vector of point P is 3i + 2j + 5k, what is the component of P along the x-axis?
A.
5
B.
2
C.
3
D.
0
Show Answer & Explanation
Correct Answer: C
The component of a vector along the x-axis is simply the coefficient of i, which is 3.
Q4
Medium
If position vector of point P is given as (2, 3, 4), what is the vector equation of the line passing through P parallel to vector A = (1, -1, 2)?
A.
r(t) = (2 + t, 3 - t, 4 + 2t)
B.
r(t) = (2 - t, 3 + t, 4 - 2t)
C.
r(t) = (2 + 2t, 3 - t, 4 + t)
D.
r(t) = (2, 3, 4) + t(1, -1, 2)
Show Answer & Explanation
Correct Answer: D
The vector equation of a line through point P parallel to vector A is given as r(t) = position + t(direction). This fits option D.
Q5
Medium
Two vectors A = 3i + j + k and B = -i + 4j - 2k are said to be coplanar. What is the condition that confirms their coplanarity?
A.
A ยท (B ร C) = 0
B.
A ร B = 0
C.
A + B = 0
D.
A ยท B = 0
Show Answer & Explanation
Correct Answer: A
The condition for coplanarity of vectors A, B, and C is that the scalar triple product A ยท (B ร C) should be zero.
Q6
Medium
The points A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9) form a vector triangle. What is the area of this triangle?
A.
0
B.
3
C.
4.5
D.
6
Show Answer & Explanation
Correct Answer: A
The area of the triangle formed by points A, B, and C can be determined using the formula. Since the points are collinear, the area is 0.
Q7
Medium
If the volume of the parallelepiped formed by vectors A, B, and C is given by the scalar triple product, what is the volume formed by vectors A = (1, 2, 3), B = (4, 5, 6), and C = (7, 8, 9)?
A.
1
B.
0
C.
2
D.
3
Show Answer & Explanation
Correct Answer: B
The volume of the parallelepiped is given by the absolute value of the scalar triple product |A ยท (B ร C)|. In this case, the points A, B, and C are coplanar, resulting in a volume of 0.
Q8
Hard
A line passes through the point (3, 2, 1) and has a direction ratio of (2, -1, 3). Find a point on this line with integer coordinates.
A.
(5, 1, 4)
B.
(4, 4, 6)
C.
(6, 5, 7)
D.
(1, 0, 0)
Show Answer & Explanation
Correct Answer: A
Using the parametric equations of the line, you can find integer points by setting the parameter appropriately.
Q9
Hard
Given points A(0, 0, 0), B(1, 1, 1), and C(1, 0, 0), determine if the triangle formed by these points lies in a single plane.
A.
Yes, it lies in a single plane.
B.
No, it does not lie in a single plane.
C.
It is a right-angled triangle.
D.
It is an equilateral triangle.
Show Answer & Explanation
Correct Answer: A
The points A, B, and C can be checked for coplanarity by examining the vectors AB and AC, and confirming they are not collinear.
Q10
Hard
The angle between the vectors a = (3, -2, 1) and b = (-1, 0, 4) is given by which of the following formulas?
A.
cos(theta) = (a . b) / (|a| |b|)
B.
sin(theta) = (a x b) / (|a| |b|)
C.
tan(theta) = (a + b) / (a - b)
D.
theta = a / b
Show Answer & Explanation
Correct Answer: A
The angle ฮธ between two vectors is determined using the dot product formula: cos(ฮธ) = (a . b) / (|a| |b|).
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Vector Algebra and 3D Geometry โ JEE Mathematics Practice Questions Online
This page contains 1678 practice MCQs for the chapter Vector Algebra and 3D Geometry in JEE Mathematics. The questions are organized by difficulty โ 384 easy, 718 medium, 576 hard โ so you can choose the right level for your preparation.
Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit. This chapter covers 3 topics, giving you comprehensive coverage of the entire chapter.