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Question :

Read each statement below carefully and state with reasons if it is true or false : (a) The magnitude of a vector is always a scalar, (b) each component of a vector is always a scalar, (c) the total path length is always equal to the magnitude of the displacement vector of a particle. (d) the average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of the average velocity of the particle over the same interval of time, (e) Three vectors not lying in a plane can never add up to give a null vector.

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Last updated date: 19th Jun 2024
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Answer
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Hint: To determine the truth or falsity of these statements, consider the definitions and properties of vectors and scalars in physics.


Step-by-Step Solutions:

(a) The magnitude of a vector is always a scalar:

Explanation: The magnitude of a vector is defined as a scalar quantity representing the length or size of the vector without any directional information. It's always a non-negative scalar value.


(b) Each component of a vector is always a scalar:

Explanation: The components of a vector are scalar values that represent the projections of the vector onto specific axes (e.g., x, y, z). Each component is a scalar.


(c) The total path length is always equal to the magnitude of the displacement vector of a particle:

Explanation: The total path length is generally greater than or equal to the magnitude of the displacement vector. The displacement vector is the straight-line distance from the initial position to the final position. The path length, on the other hand, takes into account the actual distance traveled, which may be longer if the particle moves along a curved or non-straight path.


(d) The average speed of a particle is either greater or equal to the magnitude of the average velocity of the particle over the same interval of time:

Explanation: This statement is true. Average speed is defined as the total path length divided by the time taken. Since path length is always greater than or equal to displacement, and time remains the same in both cases, the average speed is always greater than or equal to the magnitude of average velocity.


(e) Three vectors not lying in a plane can never add up to give a null vector:

Explanation: Three vectors not lying in a plane can indeed add up to give a null vector. This is known as the triangle law of vector addition. If three vectors form a closed triangle, their vector sum is the null vector. The key is that the vectors must form a closed geometric shape, and they need not lie in the same plane.


Note: Understanding the definitions and properties of vectors and scalars is essential for working with physics concepts. The difference between displacement and path length is particularly important as it relates to the concepts of speed and velocity. Also, the triangle law of vector addition is a fundamental principle in vector mathematics.

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