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Correct option is: (A) zero vector

Correct option is: (D) true only when

Correct option is: (B) displacement

1. Graphical representation:

A vector is graphically represented by a directed line segment or an arrow. 

eg.: displacement of a body from P to Q is represented as P → Q.

2. Symbolic representation:

Symbolically a vector is represented by a single letter with an arrow above it, such as \(\overset{\rightarrow}{A}\). The magnitude of the vector \(\overset{\rightarrow}{A}\) is denoted as |A| or | \(\overset{\rightarrow}{A}\) | or A.

Unit vector: A vector having unit magnitude in a given direction is called a unit vector in that direction.

If \(\overset\rightarrow{p}\) is a non zero vector (P ≠ 0) then the unit vector \(\hat{u}_p=\frac{\overset\rightarrow{P}}{P}\)

∴ \(\overset\rightarrow{p}=\hat{u}_p\,P\)

Significance of unit vector:

i. The unit vector gives the direction of a given vector.

ii. Unit vector along X, Y and Z direction of a rectangular (three dimensional) coordinate is represented by \(\hat{i}\), \(\hat{j}\) and \(\hat{K}\) respectively Such that \(\hat{u}_x=\hat{i}\), \(\hat{u}_y=\hat{j}\) and \(\hat{u}_z=\hat{K}\)

This gives \(\hat{i}=\frac{\overset\rightarrow{X}}{X}\), \(\hat{j}=\frac{\overset\rightarrow{Y}}{X}\) and \(\hat{K}=\frac{\overset\rightarrow{Z}}{Z}\)

Statement of Right handed screw rule: Hold a right handed screw with its axis perpendicular to the plane containing vectors and the screw rotated from first vector to second vector through a small angle, the direction in which the screw tip would advance is the direction of the vector product of two vectors.

1. The process of splitting a given vector into its components is called resolution of the vector.

2. Resolution of vector is equal to replacing the original vector with the sum of the component vectors.

1. Scalars: Distance travelled, speed, work done, energy. 

2. Vectors: Displacement, velocity, force.

Correct option is: (C) 26

Correct option is: (D) 8 square units

The diagonal of the parallelogram made by two vectors as adjacent sides not passing through common point of two vectors represents triangle law of vector addition.

1. For a physical quantity, only having magnitude and direction is not a sufficient condition to be a vector.

2. A physical quantity also has to obey vectors law of addition to be termed as vector.

3. Hence, anything that has magnitude and direction is not necessarily a vector.

Example: Though current has definite magnitude and direction, it is not a vector.

When two vectors of a type are inclined with each other, their resultant can be determined by using triangle law of vector addition.

If number of vectors are represented by the various sides of a closed polygon taken in one order then, their resultant is always zero.

Correct option is: (C) 90°

1. Physical quantities which can be completely described b their magnitude (a number and unit) are called scalars. 

2. Physical quantities which need magnitude, as well as direction for their complete description, are called vectors.

Correct answer is (C) same magnitude and direction

Correct answer is (D) √3/2

Correct answer is (A) zero

Correct answer is (C) \(\vec{A}\times\vec{B} = \vec{B} \times \vec{A}\)

Using the triangle law of addition of vectors,

\(\overset\longrightarrow{AC}\) + \(\overset\longrightarrow{CB}\) = \(\overset\longrightarrow{AB}\)

∴ \(\overset\longrightarrow{AC}\) = \(\overset\longrightarrow{AB}\) – \(\overset\longrightarrow{CB}\)

Scalar product of two vectors:

1. The scalar product of two non-zero vectors is defined as the product of the magnitude of the two vectors and cosine of the angle θ between the two vectors.

2. The dot sign is used between the two vectors to be multiplied therefore scalar product is also called dot product.

3. The scalar product of two vectors \(\overset\rightarrow{P}\) and \(\overset\rightarrow{Q}\) is given by, \(\overset\rightarrow{P}\). \(\overset\rightarrow{Q}\) = PQ cos θ where, p = magnitude of \(\overset\rightarrow{P}\), Q = magnitude of \(\overset\rightarrow{Q}\)

θ = angle between \(\overset\rightarrow{P}\) and \(\overset\rightarrow{Q}\)

4. Examples of scalar product:

i. Power (P) is a scalar product of force (\(\overset\rightarrow{F}\)) and velocity (\(\overset\rightarrow{V}\))

∴ P = \(\overset\rightarrow{F}\). \(\overset\rightarrow{V}\)

ii. Work is a scalar product of force (\(\overset\rightarrow{F}\)) and displacement (\(\overset\rightarrow{S}\)).

∴ W = \(\overset\rightarrow{F}\). \(\overset\rightarrow{S}\)