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Question :

State with reasons whether the following algebraic operations with scalar and vector physical quantities are meaningful :

(a) adding any two scalars, (b) adding a scalar to a vector of the same dimensions, (c) multiplying any vector by any scalar, (d) multiplying any two scalars, (e) adding any two vectors, (f) adding a component of a vector to the same vector.

Last updated date: 23rd Jun 2024
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Hint: To determine whether these operations are meaningful, consider the fundamental properties of scalar and vector quantities and how these operations relate to those properties.

Step-by-Step Solutions:

(a) Adding any two scalars:

  • Explanation: Adding two scalars is meaningful because scalars have only magnitude and no direction. You can simply add their numerical values.

(b) Adding a scalar to a vector of the same dimensions:

  • Explanation: This is meaningful because it's equivalent to scaling the vector by the scalar quantity. For example, if you add a scalar speed to a vector velocity, you are just changing the magnitude of the vector in the same direction.

(c) Multiplying any vector by any scalar:

  • Explanation: Multiplying a vector by a scalar is meaningful and represents a scaling or stretching of the vector. The direction of the vector remains the same, but its magnitude is scaled by the scalar.

(d) Multiplying any two scalars:

  • Explanation: Multiplying two scalars is also meaningful. This operation results in another scalar, representing the product of the two numerical values. It is a fundamental mathematical operation.

(e) Adding any two vectors:

  • Explanation: Adding two vectors is meaningful when they have the same dimensions and units. The result is a vector, and the addition follows the rules of vector addition, which considers both magnitude and direction.

(f) Adding a component of a vector to the same vector:

  • Explanation: This is meaningful because you are essentially adding two vectors that have the same direction. This operation is a specific case of vector addition, and the result is a vector with the same direction but an altered magnitude.

Scalar quantities represent magnitude only, while vector quantities have magnitude and direction. Operations involving scalars are straightforward mathematical operations, whereas operations involving vectors require consideration of both magnitude and direction. Adding a scalar to a vector or multiplying a vector by a scalar is meaningful and represents scaling or altering the magnitude of the vector. Adding components of vectors to the same vector is also a valid vector addition operation in which the direction remains unchanged, but the magnitude may be affected.