Question

To introduce a vector quantity .... (A) it needs magnitude not direction (B) it needs direction not magnitude (C) it needs both magnitude and direction (D) nothing is needed

Short answer

(C) A vector quantity needs both magnitude and direction.

Step-by-step solution

Understanding Vector Quantity

A vector quantity is a quantity that has both magnitude(complete size or value) and direction. To specify a vector, we must know both the magnitude (how much or the size) and the direction (where it is pointing). Examples of vector quantities are force, velocity, and acceleration. Now, let's analyze the given choices.

Choice A: Needs Magnitude, Not Direction

Since we know that a vector quantity requires both magnitude and direction, this choice is incorrect. A quantity with only magnitude, but no direction, is called a scalar quantity.

Choice B: Needs Direction, Not Magnitude

This choice is also inaccurate because a vector cannot exist without both magnitude and direction. Without magnitude, the vector would not have any value associated with it.

Choice C: Needs Both Magnitude and Direction

This choice is correct since vector quantities require both magnitude and direction. To fully define a vector, we must provide the necessary information about both its size (magnitude) and the direction in which it's pointing or acting.

Choice D: Nothing is Needed

This choice is incorrect as vector quantity cannot be defined without specifying its magnitude and direction. A quantity with neither magnitude nor direction does not exist in the context of vector quantities. To conclude, the correct choice is: (C) A vector quantity needs both magnitude and direction.

Key concepts

Understanding Magnitude

Magnitude is an essential characteristic of vector quantities. It represents the size or amount of a vector. Imagine you are measuring how strong a force is or how fast something is moving. Magnitude tells us about the 'how much' part. For instance, if a car is moving at 60 km/h, we say its speed is 60. Here, '60' is the magnitude. Magnitude is always a non-negative number. It measures the length, size, or distance related to the vector.

When you calculate the magnitude of a vector in two dimensions, say \(a= \begin{bmatrix} a_1 \ a_2 \end{bmatrix}\), you use the formula:\

• \( \text{Magnitude of } a = \sqrt{a_1^2 + a_2^2} \)

This is derived from the Pythagorean theorem, as we are essentially finding the hypotenuse of a right-angled triangle formed by the components of the vector. Magnitude provides a scalar representation of the vector's size.

Concept of Direction

Aside from magnitude, direction is the other critical component that makes a quantity a vector. While magnitude tells us how much there is of something, direction tells us where this 'something' is going or pointing. Imagine a river flowing with a certain speed; knowing just the speed (magnitude) isn't enough if we don't know the direction it flows.

Direction is usually represented using angles, such as degrees or radians in mathematical contexts. In the Cartesian coordinate system, we often use angles measured from the positive x-axis to indicate direction. A vector is defined by both its magnitude and direction, and changing either alters the vector itself. When we express direction in notation, it is done using unit vectors or directional angles. For example, the direction in a 2D vector \(b = b_1 \hat{i} + b_2 \hat{j}\) can be found using its components:

• \( \text{Direction angle } \theta = \arctan\left(\frac{b_2}{b_1}\right) \)

Here, direction not only guides the path of the vector but also influences its overall impact in vector operations.

Differences Between Scalar and Vector Quantities

While both scalar and vector quantities feature magnitude, only vector quantities also include direction, which is key to their definition. Scalar quantities are simple because they only consist of a magnitude. Examples of scalar quantities include speed, mass, and time, where direction is not involved.

In contrast, vector quantities like velocity or force require knowledge of both how much (the magnitude) and in which direction the quantity acts. This makes vector quantities a little more complex but more informative when describing natural phenomena. For visualizing the difference:

• A scalar akin to a number saying 'how much', like '5 seconds' or '100 kilograms'.
• A vector, meanwhile, is like a full sentence indicating both 'how much' and 'where', such as '10 meters towards the north'.

Therefore, understanding these differences is vital when analyzing physical phenomena or solving physics and engineering problems that involve direction-sensitive quantities.