NCERT Solutions for Class 12 Science Maths Chapter 4 – Vector Algebra
Explore detailed solutions for Class 12 Science Maths Chapter 4 on Vector Algebra, featuring clear, step-by-step explanations. Widely embraced by Class 12 Science students, these Maths Vector Algebra Solutions are invaluable for efficiently finishing homework assignments and gearing up for examinations. Free access to all questions and answers from Chapter 4 of the NCERT Book for Class 12 Science Maths is available here. Page No 428: Question 1: Represent graphically a displacement of 40 km, 30° east of north. ANSWER: Here, vector represents the displacement of 40 km, 30° East of North. Page No 428: Question 2: Classify the following measures as scalars and vectors. (i) 10 kg (ii) 2 metres north-west (iii) 40° (iv) 40 watt (v) 10–19 coulomb (vi) 20 m/s2 ANSWER: (i) 10 kg is a scalar quantity because it involves only magnitude. (ii) 2 meters north-west is a vector quantity as it involves both magnitude and direction. (iii) 40° is a scalar quantity as it involves only magnitude. (iv) 40 watts is a scalar quantity as it involves only magnitude. (v) 10–19 coulomb is a scalar quantity as it involves only magnitude. (vi) 20 m/s2 is a vector quantity as it involves magnitude as well as direction. Page No 428: Question 3: Classify the following as scalar and vector quantities. (i) time period (ii) distance (iii) force (iv) velocity (v) work done ANSWER: (i) Time period is a scalar quantity as it involves only magnitude. (ii) Distance is a scalar quantity as it involves only magnitude. (iii) Force is a vector quantity as it involves both magnitude and direction. (iv) Velocity is a vector quantity as it involves both magnitude as well as direction. (v) Work done is a scalar quantity as it involves only magnitude. Page No 428: Question 4: In Figure, identify the following vectors. (i) Coinitial (ii) Equal (iii) Collinear but not equal ANSWER: (i) Vectors and are coinitial because they have the same initial point. (ii) Vectorsandare equal because they have the same magnitude and direction. (iii) Vectorsand are collinear but not equal. This is because although they are parallel, their directions are not the same. Page No 428: Question 5: Answer the following as true or false. (i) andare collinear. (ii) Two collinear vectors are always equal in magnitude. (iii) Two vectors having same magnitude are collinear. (iv) Two collinear vectors having the same magnitude are equal. ANSWER: (i) True. Vectors andare parallel to the same line. (ii) False. Collinear vectors are those vectors that are parallel to the same line. (iii) False. It is not necessary for two vectors having the same magnitude to be parallel to the same line. (iv) False. Two vectors are said to be equal if they have the same magnitude and direction, regardless of the positions of their initial points. Page No 440: Question 1: Compute the magnitude of the following vectors: ANSWER: The given vectors are: Page No 440: Question 2: Write two different vectors having same magnitude. ANSWER: Hence, are two different vectors having the same magnitude. The vectors are different because they have different directions. Page No 440: Question 3: Write two different vectors having same direction. ANSWER: The direction cosines of are the same. Hence, the two vectors have the same direction. Page No 440: Question 4: Find the values of x and y so that the vectors are equal ANSWER: 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. Page No 440: Question 5: Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (–5, 7). ANSWER: The vector with the initial point P (2, 1) and terminal point Q (–5, 7) can be given by, Hence, the required scalar components are –7 and 6 while the vector components are Page No 440: Question 6: Find the sum of the vectors. ANSWER: The given vectors are. Page No 440: Question 7: Find the unit vector in the direction of the vector. ANSWER: The unit vector in the direction of vector is given by. Page No 440: Question 8: Find the unit vector in the direction of vector, where P and Q are the points (1, 2, 3) and (4, 5, 6), respectively. ANSWER: The given points are P (1, 2, 3) and Q (4, 5, 6). Hence, the unit vector in the direction of is . Page No 440: Question 9: For given vectors, and , find the unit vector in the direction of the vector ANSWER: The given vectors are and. Page No 440: Question 10: Find a vector in the direction of vector which has magnitude 8 units. ANSWER: Hence, the vector in the direction of vector which has magnitude 8 units is given by, Page No 440: Question 11: Show that the vectorsare collinear. ANSWER: . Hence, the given vectors are collinear. Page No 440: Question 12: Find the direction cosines of the vector ANSWER: Hence, the direction cosines of Page No 440: Question 13: Find the direction cosines of the vector joining the points A (1, 2, –3) and B (–1, –2, 1) directed from A to B. ANSWER: The given points are A (1, 2, –3) and B (–1, –2, 1). Hence, the direction cosines of are Page No 440: Question 14: Show that the vector is equally inclined to the axes OX, OY, and OZ. ANSWER: 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. Page No 440: Question 15: Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are respectively, in the ration 2:1 (i) internally (ii) externally ANSWER: The position vector of point R dividing the line segment joining two points P and Q in the ratio m: n is given by: Position vectors of P and Q are given as: (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, (ii) The …
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