CBSE Class 12 Mathematics Chapter 10: Vector Algebra NCERT Solutions

To represent the displacement graphically, we use a scale. Let's choose a scale where 10 km is represented by a certain length on paper. For example, we can use a scale of 1 cm = 10 km.

The displacement has a magnitude of 40 km and a direction of 30° east of north. This means the direction is 30° measured from the North direction towards the East direction.

We draw a line segment starting from an origin point (O). This line segment should have a length corresponding to 40 km according to our scale (e.g., 4 cm if 1 cm = 10 km). The direction of this line segment is crucial: it should make an angle of 30° with the North direction, pointing towards the East.

Let the line segment be represented by $\backslash overrightarrow\{OP\}$. The North direction is typically represented upwards, and the East direction to the right. The angle between the North direction (positive y-axis) and $\backslash overrightarrow\{OP\}$ is 30°.

Scale: 1 cm = 10 km (or any suitable scale)

Graphical Representation:

Draw a coordinate system with North pointing up and East pointing right. From the origin O, draw a vector $\backslash overrightarrow\{OP\}$ of length 4 units (representing 40 km) such that it makes an angle of 30° with the North direction (positive y-axis) towards the East.

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

1. 10 kg: This represents mass, which has only magnitude. So, it is a scalar quantity.
2. 40 watt: This represents power, which has only magnitude. So, it is a scalar quantity.
3. 40°: This represents an angle, which has only magnitude. So, it is a scalar quantity.
4. 20 m/s²: This represents acceleration. Acceleration has both magnitude (20 m/s²) and direction. Therefore, it is a vector quantity.
5. 10⁻¹⁰ Coulomb: This represents electric charge, which has only magnitude. So, it is a scalar quantity.
6. 2 meters north-west: This represents a displacement or distance with a specified direction (north-west). Since it has both magnitude (2 meters) and direction, it is a vector quantity.

We determine whether each quantity requires a direction to be fully described.

1. Time period: This is the duration of time for one cycle, which is described solely by its magnitude. Hence, it is a scalar quantity.
2. Velocity: Velocity is the rate of change of displacement, which includes both speed (magnitude) and direction. Hence, it is a vector quantity.
3. Force: Force is a push or pull that has both magnitude and a direction in which it acts. Hence, it is a vector quantity.
4. Distance: This is the total path length covered, which is described solely by its magnitude. Hence, it is a scalar quantity.
5. Work done: Work done is calculated as the product of force and displacement (in the direction of force), and it is described solely by its magnitude. Hence, it is a scalar quantity.

Based on the typical definitions and visual representation of vectors in a diagram:

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

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

(iii) Collinear but not equal vectors: These are vectors that are parallel (lie on the same line or parallel lines) but do not have the same direction or magnitude. If vectors $\backslash vec\{a\}$ and $\backslash vec\{c\}$ are parallel but point in opposite directions or have different lengths, they are collinear but not equal.

Let's analyze each statement:

(i) True. A vector $-\backslash vec\{a\}$ has the same magnitude as $\backslash vec\{a\}$ but points in the opposite direction. Vectors pointing in opposite directions are parallel and thus lie on the same line, making them collinear.

(ii) False. Collinear vectors lie on the same line or parallel lines. 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) False. Two vectors can have the same magnitude but point in completely different directions (e.g., along the x-axis and y-axis). In this case, they are not collinear.

(iv) True. If two vectors are collinear, they are parallel. If they also have the same magnitude, they must be equal because equality requires both the same magnitude and the same direction. Since they are collinear, they are either in the same direction or opposite directions. If they have the same magnitude and are in the same direction, they are equal. If they have the same magnitude and are in opposite directions, they are negatives of each other, not equal vectors (unless the magnitude is zero).