Content Curator
In physics, only two directions are allowed in one dimension. The motion of an object down a straight line involves two-directional features of quantities that can be handled by + and – signs. However, we must employ vectors to explain the motion of an object in two dimensions (a plane) or three dimensions (space). As a result, learning the language of vectors is required first.
What is a Vector?
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A vector is an object which has both magnitude and direction. Magnitude defines the size of the vector. Is represented by a line with an arrow, with the length of the line representing the vector's magnitude and the arrow indicating the direction. The Euclidean vector is also known as the Geometric vector, Spatial vector, or simply "vector."
A Vector
Since vectors help us to find the direction triangle method, the parallelogram method and the component method can help to add and find the resultant.
Resolution of Vectors
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Resolution of vectors is splitting a vector into various parts. These parts of vectors act in different directions and are called “Components of Direction”.
Resolution of Vectors
We can resolve a vector mainly around two-dimensional and three-dimensional figures.
Resolution of Vectors into Rectangular Components
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Resolution of Vectors into Rectangular Components
The component vectors of a vector are called rectangular components of a vector when they are split into two component vectors at right angles to each other.
Assume a vector is on the x-y plane and forms angles a and b with the x- and y-axes, respectively, as illustrated in the diagram.
Vector Quadrants
The components of a vector AB have different signs depending on which quadrant it is in.
Rectangular Components of Vectors in Three Dimensions
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Rectangular Components of Vectors in Three Dimensions
The position vector is the vector that connects a point P to the origin if its coordinates are (x, y, z).
The vector of point P's location is
A vector A resolved into components along x-, y- and z-axes
Triangle Law of Vector Addition
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The triangle law of vector addition asserts that if two vectors are represented as two sides of a triangle with the same order of magnitude and direction, the triangle will display the magnitude and direction of the resulting third side vector.
Triangle Law of Vector Addition
Parallelogram Law of Vector Addition
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If the opposite sides of a parallelogram formed from a location can represent two vectors acting concurrently at that point in magnitude and direction, then the diagonal of the parallelogram crossing through that point can be used to represent the resultant vector in magnitude and direction.
Parallelogram Law of Vector Addition
Resolution of Vectors: Things to Remember
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- A vector is an object which has both magnitude and direction.
- The resolution of vectors is splitting a vector into various parts. These parts of vectors act in different directions and are called “Components of Direction”.
- In Triangle Law of Vector Addition, the triangle will display the magnitude and direction of the resulting third side vector.
- In Parallelogram Law of Vector Addition, the diagonal of the parallelogram crossing through that point can be used to represent the resultant vector in magnitude and direction.
- Resolution of Vectors is covered under CBSE Class 11 Physics Unit 2 Kinematics Chapter 4 Motion in a Plane. Unit 2 carries a combined weightage of 23 marks.
Sample Questions
Ques. A vector is inclined at an angle of −300−300 to the positive x-direction, and has a length of 10 units. Specify the vector using the i^- ^j system. (3 marks)
Ans:
We have:
|||−−→AC|||=10cos300=10(√2)=5√3|||−−→CB|||
=10sin300=10(12)=5
|AC→|=10cos30o=10(32)=53|CB→|=10sin30o=10(12)=5
Thus,
−−→AB=−−→AC+−−→CB =5√3ˆi−5ˆj
Ques. 3i+2j+3k is the velocity of a particle. What is the component of the velocity vector along the line i-j+k? (3 marks)
Ans. Let V = 3 i + 2 j + 3 k is the given velocity vector.
| V | cos, where is the angle between a given velocity vector and the provided line, is the vector component of V along the line L = (i-j+k).
angle θ is calculated as
As a result, along the line (i-j+k), the vector component of V is (0.492)(3I + 2j + 3k).
Ques. Can vectors be associated with
(a) The length of a wire bent into a loop
(b) A plane area
(c) A sphere? Explain. (3 marks)
Ans. (a) We cannot associate a vector with the length of a wire bent into a loop. This is because the length of the loop does not have a definite direction.
(b) We can associate a vector with a plane area. Such a vector is called area vector and its direction is represented by a normal drawn outward to the area.
(c) The area of a sphere does not point in any definite direction. However, we can associate a null vector with the area of the sphere. We cannot associate a vector with the volume of a sphere.
Ques. A force of magnitude F is acting on a box at an angle of θ to the horizontal. Take the horizontal right direction as the ˆi^ direction, and the vertically up direction as the ˆj^direction. Specify the force →F→using this ˆii^- ˆjj^ system. (3 marks)
Ans.
We have:
−→F1=Fcosθˆi,−→F2=Fsinθˆj
F1→=Fcos?θi^,F2→=Fsin?θj^
Thus,
→F=−→F1+−→F2
=Fcosθˆi+Fsinθˆ
Ques. A vector a→is specified in the i^- j^ system as a→=3i^−5j^. Find the magnitude of a→, and its inclination with the positive x-direction. (3 marks)
Ans.
Using Pythagoras Theorem, we have:
|→a|=√32+52=√34
|a→|=32+52=34
The angle θ is given by
tanθ=53⇒θ
=tan−1(53)tanθ=53⇒θ=tan−1(53)
Thus, the angle of inclination of this vector with the horizontal is −tan−1(53)−tan−1(53). We note that in general, a vector →r=xˆi+yˆjr→=xi^+yj^ will have a magnitude of
||→r|=√x2+y2|r→|=x2+y2.
Ques. What do you understand about the components of vectors? (1 mark)
Ans: When we break vectors apart into their parts,those parts are called components.For example, in vector(4,1) the x-component is 4 and y-component is 1.
Ques. How many components does a vector have? (1 mark)
Ans:A vector with two directions in two dimensions is known as a two-dimensional vector. The component describes the vector's impact in that particular context.
Ques. Why do we resolve vectors? (2 marks)
Ans:It is most advantageous to resolve a vector into components that are perpendicular to one another, usually horizontal and vertical, while addressing physics problems.
Ques. A vector has magnitude and direction. Does it have a location in the space?
Can it vary with time?
Will two equal vectors a and b at different locations in space necessarily have identical physical effects? Give examples in support of your answer. (3 marks)
Ans.
- Besides having magnitude and direction, each vector also has a location in space.
- A vector can vary with time. As an example, velocity and acceleration vectors may vary with time.
- Two equal vectors a and b having different locations may not have the same physical effect. As an example, two balls thrown with the same force, one from earth and the other from moon will attain different ‘maximum heights’.
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