Boards Concepts Booster Class 12 Vector Algebra

BOARDS CONCEPTS BOOSTER VECTOR ALGEBRA  Download Doubtnut Today Ques No. Question CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 1. INTRODUCTION 1 1. Two type of numbers - Vector and scalar.  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 2. EVOLUTION OF VECTOR CONCEPT 2 1. Evolution of vector concept  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 2. EVOLUTION OF VECTOR CONCEPT 3 2. Representation of vector direction sense magnitude of vector or length  Click to LEARN this concept/topic on Doubtnut

CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 3. TYPES OF VECTORS 1. Equal vectors, null vector, unit vectors, position vector, like and unlike vectors,collinear and parallel vectors 4  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 3. TYPES OF VECTORS 2. Co-initial vectors, coterminous vector and co-planar vectors,negative of a vector,reciprocal vectors 5  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 3. TYPES OF VECTORS 6 3. Free vector and localized vector  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 3. TYPES OF VECTORS 4. In a regular hexagon find which vectors are collinear, equal, coinitial, collinear but not equal. 7  Click to LEARN this concept/topic on Doubtnut

CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 4. ADDITION OF VECTOR 8 1. Parallelogram law of addition of vectors  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 4. ADDITION OF VECTOR 9 2. Properties of addition of vectors  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 4. ADDITION OF VECTOR 10 3. Magnitude of addition of two vectors  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 4. ADDITION OF VECTOR 11 4. Commutative law  Click to LEARN this concept/topic on Doubtnut

CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 4. ADDITION OF VECTOR 12 5. Associative law  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 4. ADDITION OF VECTOR 13 6. Existence of additive identity  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 4. ADDITION OF VECTOR 14 7. Existence of additive inverse  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 4. ADDITION OF VECTOR 8. ABC is a triangle and P any point on BC. If show that is a parallelogram and ABQC is the sum of therefore; is a fixed point. ; 15 ¯ ¯ ¯ P Q ¯ ¯ ¯ A P +¯ ¯ ¯ P B +¯ ¯ ¯ P C Q  Click to LEARN this concept/topic on Doubtnut

CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 4. ADDITION OF VECTOR 16 9. Addition of vectors in 3 dimensional space  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 4. ADDITION OF VECTOR 10. ABCD is a parallelogram. E; F are mid point of BC; CD respectively. AE; AF meet the diagonal BD at points Q and P respectively. Show that points P and Q trisect DB. 17  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 5. COMPONENTS OF A VECTOR 18 1. Component of vector  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 6. MULTIPLICATION OF A VECTOR BY A SCALAR 19 1. Definition geometric interpretation: Multiplication Of A Vector By A Scalar  Click to LEARN this concept/topic on Doubtnut

CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 6. MULTIPLICATION OF A VECTOR BY A SCALAR 20 2. Properties of Multiplication of a vector by a scalar  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 7. POSITION VECTOR OF A POINT 21 1. Position vector and a vector in terms of position vectors of its end points  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 8. SECTION FORMULA 22 1. Section formulae (Internal division)  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 8. SECTION FORMULA 23 2. Section formula (External division)  Click to LEARN this concept/topic on Doubtnut

CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 8. SECTION FORMULA are the position vectors of A and B respectively; find the position → 24 → 3. If vector of a point C on BA produced such that BC = 3/2 BA. a b and  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 9. LINEAR COMBINATION LINEAR INDEPENDENCE AND LINEAR DEPENDENCE 25 1. Definition and physical interpretation: Linear Combination  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 9. LINEAR COMBINATION LINEAR INDEPENDENCE AND LINEAR DEPENDENCE 26 2. Linear Combination: Linear Independence And Linear Dependence  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 9. LINEAR COMBINATION LINEAR INDEPENDENCE AND LINEAR DEPENDENCE 27 3. Linearly Independent  Click to LEARN this concept/topic on Doubtnut

CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 9. LINEAR COMBINATION LINEAR INDEPENDENCE AND LINEAR DEPENDENCE 28 4. Linearly Dependent  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 9. LINEAR COMBINATION LINEAR INDEPENDENCE AND LINEAR DEPENDENCE are two non collinear vectors; then every vector → b → a → r 5. Theorem 1: If and → b → b 29 and → a → a coplanar with can be expressed in one and only one way as a linear combination: x +y .  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 9. LINEAR COMBINATION LINEAR INDEPENDENCE AND LINEAR DEPENDENCE → b → a are non coplanar vectors; then any vector → c → a → b → c can be → r 6. Theorem 2: If , and 30 expressed as linear combination: x +y +z  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 9. LINEAR COMBINATION LINEAR INDEPENDENCE AND LINEAR DEPENDENCE 31 → b → b→ → a → c → a 7. Theorem 3:If vectors , and are coplanar then det( ) = 0 c  Click to LEARN this concept/topic on Doubtnut

CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 9. LINEAR COMBINATION LINEAR INDEPENDENCE AND LINEAR DEPENDENCE 8. Examples: Prove that the segment joining the middle points of two non parallel sides of a trapezium is parallel to the parallel sides and half of their sum. 32  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 10. VECTOR JONING TWO POINTS 33 1. Position vector in 2 dimensional plane and 3 dimensional plane  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 10. VECTOR JONING TWO POINTS 34 2. Components of vector joining two point in 2 and 3 dimensional plane  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 11. DOT PRODUCT 35 1. Definition; remarks and geometrical interpretation of Scalar Product  Click to LEARN this concept/topic on Doubtnut

CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 12. PROPERTIES OF DOT PRODUCT 36 a.a = |a|2 1. and commutative and distributive law of dot product  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 12. PROPERTIES OF DOT PRODUCT 2. l→ a .m→ b = lm(→ a 37 .→ b ) → a .→ are perpendicular if → b → b → a → a and then and and are not null vector b = 0  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 12. PROPERTIES OF DOT PRODUCT 3. 2 → a ±→ 2 → a∣∣ ∣∣∣ b∣∣∣ =∣∣ 2 → b∣∣∣ +∣∣∣ → b∣∣∣cosθ ± 2∣∣→ a∣∣∣∣∣ 38 and

a +→ (→ b ) a −→ 2 .(→ → a∣∣ b ) =∣∣ 2 → b∣∣∣ −∣∣∣  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 12. PROPERTIES OF DOT PRODUCT 39 4. Dot product in terms of components  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 12. PROPERTIES OF DOT PRODUCT 5. Using dot product of vectors; prove that a parallelogram; whose diagonal are equal; is a rectangle. 40  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 13. APPLICATIONS OF DOT (SCALAR) PRODUCT 41 1. Angle between two vectors in terms of dot product  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA

13. APPLICATIONS OF DOT (SCALAR) PRODUCT 42 2. Components of vector along and perpendicular a vector  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 13. APPLICATIONS OF DOT (SCALAR) PRODUCT 43 3. Cosine rule using dot product  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 13. APPLICATIONS OF DOT (SCALAR) PRODUCT 4. Prove by vector method that cos(A + B) 44 = cosAcosB − sinAsinB  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 14. VECTOR (OR CROSS) PRODUCT OF TWO VECTORS 45 1. Definition and meaning of cross product  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 14. VECTOR (OR CROSS) PRODUCT OF TWO VECTORS 46

2. geometrical interpretation of cross product and area of parallelogram2. geometrical interpretation of cross product and area of parallelogram  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 14. VECTOR (OR CROSS) PRODUCT OF TWO VECTORS 47 3. Vector product: Determinant form  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 14. VECTOR (OR CROSS) PRODUCT OF TWO VECTORS 48 → b → a 4. Vector normal to the plane; passing through and  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 14. VECTOR (OR CROSS) PRODUCT OF TWO VECTORS 49 5. The area of parallelogram; triangle and quadrilateral in terms of cross product.  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 14. VECTOR (OR CROSS) PRODUCT OF TWO VECTORS → a ;→ 6. Lagrange identity: If two vector are any two vectors b 50

2 → a ×→ ∣∣∣ b∣∣∣ 2 → b∣∣∣ 2∣∣∣ → a∣∣ = (∣∣ 2 a .→ − (→ ) b )  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 14. VECTOR (OR CROSS) PRODUCT OF TWO VECTORS a −→ a +→ → b ) 51 (→ b ) (→ b ) = 2(→ 7. Show that x x a  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 14. VECTOR (OR CROSS) PRODUCT OF TWO VECTORS 52 8. Application of cross product trigonometric proof; sin(A+B) = sinAcosB + cosAsinB  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 15. PROPERTIES OF CROSS PRODUCT 53 1. Non commutative and distributive law of vector product  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA

16. SCALAR TRIPLE PRODUCT 54 1. Introduction and definition of Scalar triple product  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 16. SCALAR TRIPLE PRODUCT 55 2. Geometrical interpretation of Scalar triple product and volume of parallelepiped  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 16. SCALAR TRIPLE PRODUCT 56 3. Coplanarity of points  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 16. SCALAR TRIPLE PRODUCT 4. Find , for which the points a A(3, 2,1), 57 B(4, a,5), C(4, 2, − 2) and are coplanar. D(6,5, − 1)  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA

16. SCALAR TRIPLE PRODUCT 58 5. Find [l m n][a b c]  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 16. SCALAR TRIPLE PRODUCT 59 6. Volume of tetrahedron and parallelepiped  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 17. PROPERTIES OF SCALAR TRIPLE PRODUCT 1. Property 1: Position of the dot and cross can be interchanged without altering the product. 60  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 17. PROPERTIES OF SCALAR TRIPLE PRODUCT 2. Property 2: → b [k→ → c] 61 a → b = k[→ → c] a  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 17. PROPERTIES OF SCALAR TRIPLE PRODUCT

3. Property 3&4: a +→ → d] [→ → c b 62 → d] = [→ → c a + [→ → d] → b and right and left handed system c  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 17. PROPERTIES OF SCALAR TRIPLE PRODUCT 4. Property 5-7: Box product of coplanar vector is 0 and box product of → a ,→ a and → b 63 is 0 and box product in terms of determinant.  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 17. PROPERTIES OF SCALAR TRIPLE PRODUCT 64 5. Property 8: Distributivity of vector product over vector addition  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 17. PROPERTIES OF SCALAR TRIPLE PRODUCT → a ,→ b ,→ 6. For any three vectors a +→ b find c 65 → b +→ [→ a] → c +→ c  Click to LEARN this concept/topic on Doubtnut

CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 17. PROPERTIES OF SCALAR TRIPLE PRODUCT 7. Simplify a −→ 66 → b −→ [→ a] → c −→ b c  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 17. PROPERTIES OF SCALAR TRIPLE PRODUCT 67 8. Evaluate . Also, interpret it geometrically. [i j k]  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 18. VECTOR TRIPLE PRODUT 1. Prove that ˆi × (→ a ×→ i ) +ˆj a ×→ × (→ j ) +ˆk 68 a ×→ × (→ k ) = 2→ a  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA

18. VECTOR TRIPLE PRODUT 69 → b ).(→ → d ) =→ a .(→ → d )) (→ (→ 2. Lagrange identity: x x x x a c b c  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 19. MISCELLANEOUS 1. Find ˆi × (→ a ×ˆi) +ˆj 70 × (→ a ×ˆj) +ˆk × (→ a ×ˆk)  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 19. MISCELLANEOUS → a ,→ → c 2. Let and be any three vectors; then prove that b 71 2 a ×→ → b ×→ → b [→ a] = [→ → c] → c ×→ b c a  Click to LEARN this concept/topic on Doubtnut

CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 19. MISCELLANEOUS → a ,→ are three non-coplanar vectors and r is any arbitrary vector. Prove → c 3. that [→ and b → r]→ → c a b → r]→ 72 + [→ → a b c → r]→ → b + [→ c a → c]→ → b = [→ r a  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 19. MISCELLANEOUS 4. Find the volume of a parallelopiped having three coterminus vectors of equal ∣∣ a∣∣ θ 73 → magnitude and equal inclination with each other.  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 20. SUBTRACTION OF VECTORS 74 1. Subtraction of vectors  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 20. SUBTRACTION OF VECTORS 2. If sum of two unit vectors is a unit vector; prove that the magnitude of their difference is √3 75  Click to LEARN this concept/topic on Doubtnut

CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 20. SUBTRACTION OF VECTORS 3. Let O be the centre of a regular hexagon ABCDEF. Find the sum of the vectors , and OB, OC,OD, OE OF 76 OA  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 21. DIRECTION COSINES AND DIRECTION RATIOS 77 1. Direction cosines  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 21. DIRECTION COSINES AND DIRECTION RATIOS 78 2. Direction ratios  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 21. DIRECTION COSINES AND DIRECTION RATIOS 79 3. Direction cosines of of a vector  Click to LEARN this concept/topic on Doubtnut

CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 21. DIRECTION COSINES AND DIRECTION RATIOS 80 4. Direction ratio of the line segment joining two points  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 21. DIRECTION COSINES AND DIRECTION RATIOS 5. Two parallel vectors have proportional direction ratios and representation of vector in terms of direction ratios. r 81  Click to LEARN this concept/topic on Doubtnut CONCEPT FOR BOARDS || Chapter VECTOR ALGEBRA 21. DIRECTION COSINES AND DIRECTION RATIOS 82 6. Projection of vector along coordinate axes; if direction cosine is given  Click to LEARN this concept/topic on Doubtnut  Download Doubtnut to Ask Any Math Question By just a click  Get A Video Solution For Free in Seconds  Doubtnut Has More Than 1 Lakh Video Solutions  Free Video Solutions of NCERT, RD Sharma, RS Aggarwal, Cengage (G.Tewani), Resonance DPP, Allen, Bansal, FIITJEE, Akash, Narayana, VidyaMandir  Download Doubtnut Today

Boards Concepts Booster Class 12 Vector Algebra