Vectors are mathematical objects that have both magnitude and direction. They are used in physics and engineering to represent quantities such as force, velocity, and displacement.
Mathematics Notes for Class 12 Chapter 7
The vector, whose initial point is the origin Oand whose terminal point is P i.e., OP is called: Position vector
The dot product of unit vector j with unit vector k is: 0
If a & b are two non-zero vectors then a x b=a & b are two non-zero vectors then a x b=: –a
If i is a unit vector then i x i =: 0
For two vectors k & i, we have: k.i=: 0
For any pt. P(x,y) the position vector of P is: x
[i j k] is equal to: 1
If K& i are two unit vectors thenkx i=: j
If the angle: Direction cosines
For two vectors a&b :a.b=: ab Cos
Two non-zero vectorsa & b are parallel ifa x b=: –bxa
Two vectors a and b are perpendicular if: a . b = 0
Magnitude of the vector xî+y? is: ?x2+y2
Two non-zero vectorsu&v are perpendicular if and only ifu.v=: 0
Sine of the angle between any two non-zero vectors a & b is: | axb
Dot product of two vectors is: Commutative
If a & b are vectors along two adjacent sides of a parallelogram then area of a parallelogram is: |axb|
Which of the following is a vector quantity: Displacement
The direction cosine of x-axis are: (1, 0, 0)
A physical quantity that can be specified by a number along with unit is called: Scalar
Two vectors are equal if they: Have equal magnitude and have same direction
The dot product of unit vector i with unit vector j is: 0
Two non-zero vectors u and v are perpendicula if and only if u.v is equal to: 2
j . k x i equals: 1
2i x 2j . k =: 4
The dot product of unit vector j with unit of vector i is: 0
Moment of a force about a point is: Vector quantity
The dot product of unit vector k with unit vector k is: 1
For two vectors a&b :a.b=: b.a
For two vectors j& k, we have: j.k=: 0
If cos ?,cos ? ,cos ? are direction cosines of a vector athen: Cos2 ? + Cos2 ? + Cos2 ? = 1
Velocity, acceleration, force, weight are example of: Vectors
Two vectors are said to be equal, if they have _______ magnitude and ________ direction: Same , same
Ifu=a1i + b1j +c1k & v =a2i +b2j + c2k be two non-zero vectors then u.v=: a1a2 + b1b2 + c1c2
The null or zero vector in R3 is O =: [0,0,0]
Work done of a force on a particle: Scalar quantity
The magnitude of vector is also called its: Norm
Two vectors are said to be negative of each other if they have _______ magnitude but _________ direction: Same, Opposite
i . (j x k) is equal to: 1
The cross product or vector product of two vectors is defined: Only in space
If a & b are vectors along two adjacent sides of a triangle then area of triangle is: 1/2 |a x b|
Commutative law holds in: Inner product
If j & K are two unit vectors then jx k =: i
For three vectors a ,b,&c : (a+b) + c =: a+(b+c)
A vector v is _________ to a vector kv, where k is a scalar: Parallel
Moment of force F ? about a point is given by: Cross prouduct
Which of triples can be direction angles of a single vector =: 45
The dot product of unit vector k with unit vector i is: 0
i . i =: 1
Time, density, temperature, length, volume, speed and work are example of: Scalars
The vector i = [1, 0, 0] is called unit vector along: x-axis
For two vectors i & j, we have: i.j=: 0
For a unit vectori, i.i=: 1
The non-zero vectors a and b are parallel of a x b =: 0
j x k =: i
If a x b =0 and a . b = then: Either a=0 or b=0
The vectors lying in the same plane are called: Perpendicular vectors
[k i j] =: 1
Two vector are ________ if and only if they are non-zero scalar multiple of each other: Parallel
2i x j – k =: 2
If i & j are two unit vectors then i x j=: k
Two zero vectors isregarded to the perpendicular to: Every vector
The vector k = [0, 0, 1] is called unit vector along:z-axis
The dot product of unit vector k with unit vector j is: 0
The dot product of unit vector i with unit vector i is: 1
A scalar quanity, is one that possesses only: Magnitude
For a unit vector j , j.j= 1
If the vector 2i + 4j – yk and 2i + 6j + xk are perpendicular then x =: 4
Ifa,b & c are three vectors then vector triple product of these vectors is: axbxc
A physical quantity that posses both magnitude and direction is called: Vector
The vector i = [1,0] is called ___________ along x-axis: Unit vector
The unit vector along z-axis is denoted by: k?
If U = 2i + 7j + 9b, then U x U =: 0
j . k x i is equal to: 1
The magnitude of dot and cross product of two vectors are zero and 1 respectively, then angle between vectors is: 90
For a unit vector: 1
If a & b are two non-zero vectors, then a vector perpendicular to each of the vectors is: a x b
If any two vectors of scalar triple product are equal, then its value is: 0
For two vectors a: b+a
The sum of squares of direction angles is equal to: 1
For a unit vectori, i.i=: 1
The difference between the position vectors of two points is called the: Displacement
For a scalar triple product the position of dot and cross: Can be interchanged
Two non-zero vectorsa & b are parallel ifa x b=: –bxa
Magnitude of the vector V = [3,-4]: 5
A vector whose magnitude is unity called: Unit vector
A unit vector vector is defined as a vector whose magnitude is: Unity
The non-negative inequalities are called: Constant
A point of a solution region where two of its boundary lines intersects is called a ________ point of the solution region: Corner
Convert a linear programming problem to a mathematical form by using: Variables
To find the optimal solution evaluate the objective function at: Corner points
ax+by+c ? 0 is an: Inequality
A set consisting of all the feasible solutions of the system of linear in equalities is called: Feasible Solution set
For different values of k, the equation 4x + 5y = k represents lines ________ to the line 4x + 5y = 0: Adding
If equations of sides of the triangle are given, then intersection of any two lines give a: Vertex of the triangle
The graph of linear equation of the form ax + by = c is a line, which divides the plane into _________ disjoint regions, where a, b and c are constants and a, b are not both zero: Two
The graph of 2x + y > 2 is the open half plane the region is ________ the origin side of 2x + y = 2: Not on
If a plane passes through the vertex of cone then the intersection is: a point circle
Solution of inequality x + 2y < 6 is:(1, 1)
If the solution region is restricted to the 1st quadrant then it is called: Feasible region
Important : Review all previous years’ 12 Class Past Papers for comprehensive practice and better exam preparation.