JEE Exam  >  JEE Notes  >  Mathematics (Maths) Class 12  >  NCERT Solutions: Exercise - 10.2 - Vector Algebra

NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra

Q6: Compute the magnitude of the following vectors:
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra
Ans: The given vectors are:
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra

Q7: Write two different vectors having same magnitude.
Ans:
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra

Q8: Write two different vectors having same direction.
Ans: 
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra

Q9: Find the values of x and y so that the vectors NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra are equal.
Ans: The two vectors will be equal if their corresponding components are equal.
Hence, the required values of x and y are 2 and 3 respectively.

Q10: Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (–5, 7).
Ans: The vector with the initial point P (2, 1) and terminal point Q (–5, 7) can be given by,
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra
Hence, the required scalar components are –7 and 6 while the vector components are

Q11: Find the sum of the vectors   NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra

Q12: Find the unit vector in the direction of the vector  NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra
Ans: The unit vector  in the direction of vector  is given by .
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra

Q13: Find the unit vector in the direction of vector , where P and Q are the points (1, 2, 3) and (4, 5, 6), respectively.
Ans: The given points are P (1, 2, 3) and Q (4, 5, 6).
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra

Q14: For given vectors,  and  , find the unit vector in the direction of the vector 
Ans:
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra

Q15: Find a vector in the direction of vector  NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra
Ans:
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra

Q16: Show that the vectorsNCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra  are collinear.
Ans: 
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra

Q17: Find the direction cosines of the vector NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra
Ans:
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra

Q18: Find the direction cosines of the vector joining the points A (1, 2, –3) and B (–1, –2, 1) directed from A to B.
Ans: The given points are A (1, 2, –3) and B (–1, –2, 1).
 NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra

Q19: Show that the vector  is equally inclined to the axes OX, OY, and OZ.
Ans: 
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra
Therefore, the direction cosines of
Now, let α, β, and γbe the angles formed by  with the positive directions of x, y, and z axes.
Then, we have
Hence, the given vector is equally inclined to axes OX, OY, and OZ.

Q20: Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are  NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebrarespectively, in the ration 2:1
Ans: The position vector of point R dividing the line segment joining two points
P and Q in the ratio m: n is given by:
        i.  Internally:
    NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra
(i) The position vector of point R which divides the line joining two points P and Q internally in the ratio 2:1 is given by,
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra
(ii) The position vector of point R which divides the line joining two points P and Q externally in the ratio 2:1 is given by
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra

Q21: Find the position vector of the mid point of the vector joining the points P (2, 3, 4) and Q (4, 1, – 2).
Ans: The position vector of mid-point R of the vector joining points P (2, 3, 4) and Q (4, 1, – 2) is given by,
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra

Q22: Show that the points A, B and C with position vectors NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra, NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra respectively form the vertices of a right angled triangle. 
Ans: Position vectors of points A, B, and C are respectively given as
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra

Q23: In triangle ABC which of the following is not true:
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra  NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra
Ans: On applying the triangle law of addition in the given triangle, we have:
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra
Hence, the equation given in alternative C is incorrect.
The correct answer is C.

Q24: If  are two collinear vectors, then which of the following are incorrect
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra
C. the respective components of  are proportional
D. both the vectors  have same direction, but different magnitudes
Ans: If  are two collinear vectors, then they are parallel.
NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra
Thus, the respective components of  are proportional.
However, vectors  can have different directions.
Hence, the statement given in D is incorrect.
The correct answer is D.

The document NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra is a part of the JEE Course Mathematics (Maths) Class 12.
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FAQs on NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra

1. What are vectors in algebra?
Ans. Vectors in algebra are quantities that have both magnitude and direction, and are represented by arrows in space. They are used to describe physical quantities such as force, velocity, and displacement.
2. How do you add vectors algebraically?
Ans. To add vectors algebraically, you can use the parallelogram law or the triangle rule. In the parallelogram law, you draw two vectors as adjacent sides of a parallelogram, and the resultant vector is the diagonal of the parallelogram. In the triangle rule, you place the tail of one vector at the head of the other, and the resultant vector is the vector from the tail of the first vector to the head of the second vector.
3. What is the dot product of two vectors?
Ans. The dot product of two vectors is a scalar quantity that is calculated by multiplying the magnitudes of the two vectors and the cosine of the angle between them. It is denoted by a · b = |a| |b| cosθ, where a and b are the vectors, |a| and |b| are their magnitudes, and θ is the angle between them.
4. How do you find the magnitude of a vector?
Ans. To find the magnitude of a vector, you can use the formula |a| = √(a1^2 + a2^2 + a3^2), where a1, a2, and a3 are the components of the vector in three-dimensional space. This formula calculates the length of the vector from its components.
5. What is the cross product of two vectors?
Ans. The cross product of two vectors is a vector that is perpendicular to the plane formed by the two vectors. It is calculated by taking the determinant of a matrix formed by the unit vectors i, j, and k and the components of the two vectors. The result is a vector that is orthogonal to both of the original vectors.
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