CBSE Class 12 Mathematics Chapter 10: Vectors NCERT Solutions

To represent the displacement graphically, we first establish a coordinate system where North is typically upwards, South downwards, East to the right, and West to the left. We choose a suitable scale, for instance, 1 cm representing 10 km. A displacement of 40 km would then be represented by a line segment of length 4 cm. The direction is 30° east of north, which means starting from the North direction and moving 30° towards the East. We draw a line segment OP starting from the origin O, making an angle of 30° with the North direction towards the East, with a length corresponding to 40 km according to the chosen scale.

Scale: 1 cm = 10 km

Vector Representation:

Imagine a point O as the origin. Draw a line segment representing North upwards. From O, draw a line segment OP such that it makes an angle of 30° with the North line, towards the East. The length of OP should represent 40 km. This vector OP graphically represents the given displacement.

Here, vector $\vec{OP}$ represents the displacement of 40 km, 30° East of North.

We classify each measure based on whether it has only magnitude (scalar) or both magnitude and direction (vector).

(i) 10 kg: This represents mass. Mass has only magnitude and no direction. Therefore, 10 kg is a scalar quantity.

(ii) 2 metres north-west: This represents a displacement or distance with a specified direction (north-west). Since it has both magnitude (2 metres) and direction, it is a vector quantity.

(iii) 40°: This typically represents an angle. An angle is defined by its measure and has no inherent direction in the context of vector classification. Therefore, 40° is a scalar quantity.

(iv) 40 watt: This represents power. Power is a measure of the rate at which energy is transferred or converted, and it only has magnitude. Therefore, 40 watt is a scalar quantity.

(v) $10^{-19}$ coulomb: This represents electric charge. Electric charge is a fundamental property of matter and has only magnitude. Therefore, $10^{-19}$ coulomb is a scalar quantity.

(vi) 20 m/s²: This represents acceleration. Acceleration has both a magnitude (20 m/s²) and a direction. Therefore, 20 m/s² is a vector quantity.

We classify each quantity based on whether it possesses direction along with magnitude.

(i) Time period: Time period is the duration of time for one cycle, oscillation, or revolution. It is measured solely by its magnitude (e.g., 5 seconds). Therefore, time period is a scalar quantity.

(ii) Distance: Distance is the total length of the path covered by a moving object. It is a measure of length and does not specify direction. Therefore, distance is a scalar quantity.

(iii) Force: Force is an interaction that, when unopposed, will change the motion of an object. It has both a magnitude (how strong the push or pull is) and a direction (in which the push or pull is applied). Therefore, force is a vector quantity.

(iv) Velocity: Velocity is the rate of change of an object's position. It specifies both the speed (magnitude) and the direction of motion. Therefore, velocity is a vector quantity.

(v) Work done: Work done in physics is defined as the energy transferred when a force moves an object. Although force and displacement are vectors, their product (dot product) results in a scalar quantity representing the energy transferred. Therefore, work done is a scalar quantity.

Assuming the figure shows vectors originating from different points and having different directions and lengths, we identify them as follows:

(i) Coinitial vectors: These are vectors that have the same initial point (starting point). If vectors $\vec{a}$ and $\vec{d}$ in the figure start from the same point, they are coinitial.

(ii) Equal vectors: These are vectors that have the same magnitude (length) and the same direction. If vectors $\vec{b}$ and $\vec{d}$ have the same length and point in the exact same direction, they are equal vectors.

(iii) Collinear but not equal vectors: Collinear vectors are vectors that are parallel to each other, meaning they lie on the same line or parallel lines. They can point in the same direction or opposite directions. If vectors $\vec{a}$ and $\vec{c}$ are parallel but point in opposite directions, or if they point in the same direction but have different magnitudes, they would be collinear but not equal. For example, if $\vec{a}$ and $\vec{c}$ are parallel and point in the same direction but have different lengths, they are collinear but not equal. If they are parallel and point in opposite directions, they are collinear but not equal (unless one is the negative of the other and they have different magnitudes).

Note: The specific identification depends on the visual representation in the figure, which is not provided here. The answer assumes a typical representation where vectors $\vec{a}$, $\vec{b}$, $\vec{c}$, and $\vec{d}$ are depicted with varying lengths and directions.

Let's analyze each statement:

(i) $\vec{a}$ and $-\vec{a}$ are collinear.

True. The vector $-\vec{a}$ has the same magnitude as $\vec{a}$ but points in the exact opposite direction. Vectors pointing in opposite directions are considered collinear because they lie on the same line or parallel lines.

(ii) Two collinear vectors are always equal in magnitude.

False. Collinear vectors only need to be parallel (or anti-parallel). They can have different magnitudes. For example, a vector of length 5 units and a vector of length 10 units pointing in the same direction are collinear but not equal in magnitude.

(iii) Two vectors having same magnitude are collinear.

False. Two vectors can have the same magnitude but point in completely different directions (e.g., perpendicular directions). Collinearity requires the vectors to be parallel or anti-parallel.

(iv) Two collinear vectors having the same magnitude are equal.

False. While they have the same magnitude, collinear vectors can point in opposite directions. If they point in opposite directions, they are not equal. They are only equal if they are collinear, have the same magnitude, AND point in the same direction.