Anand Classes offers clear and comprehensive NCERT Solutions for Exercise 10.1 of Chapter 10 Vector Algebra for Class 12 Mathematics, helping students understand the basics of vectors, direction ratios, and fundamental operations with step-by-step explanations. These notes are prepared according to the latest NCERT and CBSE guidelines, making them ideal for revision, concept building, and scoring high in board exams. Click the print button to download study material and notes.
NCERT Question.1 : Represent graphically a displacement of 40 km, 30ยฐ east of north.
Solution :
Let the displacement vector be $\overrightarrow{OA}$ such that
$$|\overrightarrow{OA}| = 40$$
Displacement $40$km, $30^\circ$ East of North means the vector $\overrightarrow{OA}$ makes an angle $30^\circ$ with North in the EastโNorth quadrant.
Note
โ$\theta^\circ$ South of Westโโ means a vector in the SouthโWest quadrant making an angle of $\theta^\circ$ with West.
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NCERT Question.2 : Check the following measures as scalars and vectors.
(i) 10 kg (ii) 2 meters north-west (iii) 40ยฐ (iv) 40 Watt (v) 10โ 19 coulomb (vi) 20 m/sec2.
Solution.
(i) 10 kg is a measure of mass and therefore a scalar (10 kg has no direction; it is magnitude only).
(ii) 2 meters North-West is a measure of displacement (has magnitude and direction both) and hence is a vector.
(iii) 40ยฐ is a measure of angle, i.e., magnitude only, and therefore a scalar.
(iv) 40 Watt is a measure of power (has no direction) and therefore a scalar.
(v) $10^{-19}$ coulomb is a measure of electric charge (is magnitude only) and therefore a scalar.
(vi) 20 m/sec2 is a measure of acceleration (rate of change of velocity) and hence is a vector.
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NCERT Question.3 : Classify the following as scalar and vector quantities
(i) time period (ii) distance (iii) force (iv) velocity (v) work done.
Solution :
(i) Time period โ scalar
(ii) Distance โ scalar
(iii) Force โ vector
(iv) Velocity โ vector
(v) Work done โ scalar
(i) Time Period: Scalar (it measures duration and does not involve a direction).
(ii) Distance : Scalar (it measures the length of a path between two points and is directionless).
(iii) Force: Vector (it is described by both magnitude and the direction in which it acts).
(iv) Velocity: Vector (it represents the rate of change of position and includes direction).
(v) Work Done: Scalar (it is the energy transferred when a force is applied over a distance, but it does not inherently have a directional component).
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NCERT Question.4 : In the adjoining figure (a square), identify the following vectors
(i) co-initial
(ii) equal
(iii) collinear but not equal
Solution :
- $\vec{a}$ and $\vec{d}$ have the same initial point and therefore are co-initial vectors.
- $\vec{b}$ and $\vec{d}$ have the same direction and the same magnitude. Therefore, $\vec{b}$ and $\vec{d}$ are equal vectors.
- $\vec{a}$ and $\vec{c}$ have parallel supports, so they are collinear. Since they have opposite directions, they are not equal. Hence, $\vec{a}$ and $\vec{c}$ are collinear but not equal.
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NCERT Question.5 : Answer the following as true or false
(i) $\vec{a}$ and $-\vec{a}$ are collinear.
(ii) Two collinear vectors are always equal in magnitude.
(iii) Two vectors having same magnitude are collinear.
(iv) Two collinear vectors having the same magnitude are equal.
Solution :
(i) True
$\vec{a}$ and $-\vec{a}$ are collinear because they lie on the same line and these vectors have same magnitude but opposite in direction.
Collinear vectors are vectors that are parallel to the same line or lie on the same line, meaning they point in the same or opposite directions. Two vectors are collinear if one is a scalar multiple of the other, i.e., $\vec{a}=\lambda\vec{b}$ where ฮป is a scalar. Here ฮป =-1.
(ii) False
Collinear vectors must be aligned along the same line, but they can have different magnitudes.
$\vec{A}$ and $2\vec{A}$ are collinear but
$$|\vec{A}| \neq 2|\vec{A}|$$
so their magnitudes are not always equal.
(iii) False
Having the same magnitude does not imply that vectors are collinear.
$$|\hat{i}| = |\hat{j}| = 1$$
but $\hat{i}$ is along the xโaxis and $\hat{j}$ is along the yโaxis.
They are not collinear.
(iv) False
This statement is not necessarily true because two collinear vectors of the same magnitude can point in opposite directions.
Vectors $\vec{A}$ and $-\vec{A}$ have the same magnitude but opposite directions.
Hence, they are not equal.
Note :
Two vectors $\vec{A}$ and $\vec{B}$ are equal if they have same magnitude and same direction, that is :
(i) $|\vec{A}| = |\vec{B}|$
(ii) They have the same direction.
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