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Vectors & Three Dimensional Geometry Test - 1 - Class 12

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CBSE TEST PAPER-01 
CLASS - XII MATHEMATICS (Vectors & Three Dimensional Geometry) 
Topic: - Vectors 
1. Is the measure of 5 seconds is scalar or vector? [1] 
2. Find the sum of the vectors.
? ?
2 ,     2 4 5  6 7 a i j k b i j k c i j k = - + = - + + = - -         
?
[1] 
3. Find the direction ratios and the direction cosines of the vector
? ?
 2 7 3 r i j k = - - 
?
[1] 
4.
Find the angle between vectors    3, 2     . 6 a and b if a b a b = = =
     
[1] 
5. 
Vectors   a and b
 
 be such that 
2
3,  ,
3
a and b = =
 
then a b ×
 
 is a unit vector. 
Find angle between  a and b
 
. 
[1] 
6. Find the unit vector in the direction of the sum of the vectors
? ? ? ?
2 2 5 ,  2 3 a i j k b i j k = + - = + +
 
? ?
[4] 
7.
Show that the points 
? ?
( )
? ?
( )
? ?
( )
2 , 3 5 , 3 4 4 A i j k B i j k C i j k - + - - - - ? ? ?
 are the
vertices of right angled triangle. 
[4] 
8.
Show that the points 
? ?
( )
? ?
( )
?
( )
2 3 5 , 2 3 7 A i j k B i j k and C i k - + + + + - ? ? ?
 are
collinear. 
[4] 
9.
If , , a b c
  
are unit vector such that 0 a b c + + =
  
find the value of . . . a b b c c a + +
     
[4] 
10.
If 
? ? ? ? ?
2 2 3 ,  2  , 3 a i j k b i j k c i j = + + = - + + = +
  
? ? ?
are such that   a b is to c ? + ?
  
 is 
then find the value of ? .
[4] 
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FAQs on Vectors & Three Dimensional Geometry Test - 1 - Class 12

1. What are vectors in three-dimensional geometry?
Ans. Vectors in three-dimensional geometry are mathematical quantities that have both magnitude and direction. They are represented by a line segment in space and are commonly used to describe physical quantities such as displacement, velocity, and force in three-dimensional space.
2. How do you find the magnitude of a vector in three-dimensional geometry?
Ans. To find the magnitude of a vector in three-dimensional geometry, we use the Pythagorean theorem. If the vector is represented by (a, b, c), the magnitude can be calculated as √(a^2 + b^2 + c^2). This formula gives us the length or size of the vector in three-dimensional space.
3. What is the dot product of two vectors in three-dimensional geometry?
Ans. The dot product of two vectors in three-dimensional geometry is a scalar quantity that represents the projection of one vector onto the other. It is calculated by multiplying the corresponding components of the vectors and then summing them up. The formula for the dot product of two vectors A = (a1, a2, a3) and B = (b1, b2, b3) is A · B = a1b1 + a2b2 + a3b3.
4. How do you find the cross product of two vectors in three-dimensional geometry?
Ans. The cross product of two vectors in three-dimensional geometry is a vector that is perpendicular to both of the original vectors. It is calculated by taking the determinant of a 3x3 matrix formed by the components of the vectors. If the vectors are A = (a1, a2, a3) and B = (b1, b2, b3), the cross product is given by the vector C = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1).
5. How can vectors be applied in three-dimensional geometry?
Ans. Vectors in three-dimensional geometry have various applications. They can be used to represent the motion of objects in three-dimensional space, calculate distances and angles between points, solve problems related to forces and work, and determine the intersection of lines and planes. Vectors provide a powerful tool for analyzing and solving problems in three-dimensional geometry.
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