Physics · Class 11 Science
Ch 3Motion in a Plane — Class 11 Physics, concept-first.
When motion was confined to a straight line, direction could be captured with nothing more than a plus or minus sign. Once a particle is free to move anywhere in a plane, a single sign is no longer enough — direction itself becomes a genuinely two-dimensional idea, and every physical quantity we work with falls into on…
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Vector Equality
When are two vectors the same vector? A vector carries only two pieces of information — magnitude (length) and direction. It does not carry a fixed starting point.
Most relevant Q&A
In previous exams
How often this chapter’s concepts have been examined — real appearance data, never estimated.
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Scalars and Vectors — Physical Quantities in a Plane
When motion was confined to a straight line, direction could be captured with nothing more than a plus or minus sign.
Position and Displacement Vectors
To describe the location of a particle moving in a plane, first choose a fixed reference point , called the origin, and a pair of mutually perpendicular axes through it (the x-axis and the y-axis).
Equality of Vectors
Two vectors are said to be equal if, and only if, they have the same magnitude and point in exactly the same direction.
Multiplication of a Vector by a Real Number
A vector can be multiplied by an ordinary real number (called a scalar in this context) to produce a new vector . The rules governing this scalar multiplication are:
Addition and Subtraction of Vectors — Graphical Method
Because vectors carry direction as well as magnitude, they cannot, in general, be added by simply adding their magnitudes.
Resolution of a Vector — Rectangular and Non-Rectangular Components
Just as two vectors can be combined into a single resultant vector, a single vector can be split, or resolved, into two (or more) component vectors along chosen directions, such that the vector sum of…
Unit Vectors and Vector Addition — Analytical Method
A unit vector along any direction is a vector of magnitude exactly 1, pointing in that direction, and carrying no physical unit of its own — it exists purely to specify "which way." The unit vectors a…
Scalar (Dot) Product of Two Vectors
There are two distinct ways to "multiply" two vectors together, and they produce results of entirely different character.
Vector (Cross) Product of Two Vectors
The second way of multiplying two vectors, the vector product (or cross product), of and is defined as where is the angle between and , and is a unit vector perpendicular to the plane containing both…
Relative Velocity in a Plane
When two objects are both moving, it is often useful to ask: "how does the motion of one object appear to an observer riding along with the other?" The answer is the relative velocity, and — because v…
Motion in a Plane with Uniform Velocity and Uniform Acceleration
The equations of motion developed for a straight line extend directly to two dimensions once they are written as vector equations.
Projectile Motion
Projectile motion is the motion of an object launched into the air with some initial velocity and then left to move freely under gravity alone, air resistance being neglected.
Uniform Circular Motion
Uniform circular motion is motion in which a particle travels along a circular path of fixed radius at a constant speed .
Summary
Vectors — the basics. A scalar has magnitude only; a vector has magnitude and direction. Two vectors are equal only if both magnitude and direction match.
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
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- Q1Find the angle between the two vectors A⃗ = î - 2ĵ + 3k̂ and B⃗ = 2î + ĵ + 4k̂. **OR** The distance-time graph of a moving particle is given…Preview
- Q2a) A particle is projected with an initial velocity u, making an angle θ with the horizontal. Find the equation of the trajectory of the par…Preview
More questions
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- Example 1A vector $\vec P$ has magnitude 5 units and points in a direction making an angle of $37^\circ$ with the x-axis. A second vector $\vec Q$ ha…Free
- Example 2Two forces of magnitude 5 N and 5 N act at a point, with an angle of $60^\circ$ between them. Using the triangle law of vector addition, fin…Free
- Example 3A displacement vector of magnitude 10 m makes an angle of $30^\circ$ with the positive x-axis. Find its rectangular components $A_x$ and $A_…Free
- Example 4A vector is given as $\vec A = 6\hat i - 8\hat j$. Find (a) the magnitude of $\vec A$ and (b) a unit vector in the direction of $\vec A$.Preview
- Example 5Find the angle between the vectors $\vec A = 2\hat i$ and $\vec B = 2\hat i + 2\hat j$ using the scalar (dot) product.Preview
- Example 6Two vectors are given as $\vec A = 3\hat i$ and $\vec B = 4\hat j$. Find the vector (cross) product $\vec A \times \vec B$ and hence the are…Preview
- Example 7A boat moves with a velocity of 5 m/s due east relative to the water. The river flows at 3 m/s due north relative to the ground. Find the ma…Preview
- Example 8A ball is projected with a speed of 20 m/s at an angle of $30^\circ$ above the horizontal. Taking $g = 10\ \text{m/s}^2$ and neglecting air…Preview
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- Q9Distinguish between scalar and vector quantities, giving two examples of each. Why can ordinary algebraic addition be used for scalars but n…Free
- Q10Define the position vector and the displacement vector of a particle. If a particle's position vectors at two instants are $\vec r_1$ and $\…Free
- Q11A vector $\vec A$ is multiplied by (i) a positive scalar $\lambda > 1$, (ii) a positive scalar $0 < \lambda < 1$, and (iii) a negative scala…Free
- Q12Using the triangle law of vector addition, show that vector addition is both commutative ($\vec A + \vec B = \vec B + \vec A$) and associati…Preview
- Q13Explain the physical significance of the scalar (dot) product of two vectors, using work done by a constant force as an example. Why is work…Preview
- Q14Explain the physical significance of the vector (cross) product of two vectors, using the torque produced by a force as an example. Why must…Preview
- Q15Two vectors have magnitudes $A$ and $B$. Show, using the expression for the magnitude of their resultant, that the resultant is maximum when…Preview
- Q16Starting from $\vec A = A_x\hat i + A_y\hat j$ and $\vec B = B_x\hat i + B_y\hat j$, and using the facts $\hat i \cdot \hat i = \hat j \cdot…Preview
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- Q17Two vectors $\vec P$ (6 units) and $\vec Q$ (8 units) act at a point at right angles to each other. Find the magnitude of the resultant and…Free
- Q18A vector of magnitude 20 units makes an angle of $60^\circ$ with the positive x-axis. Find its x- and y-components.Free
- Q19For $\vec A = 5\hat i + 12\hat j$ and $\vec B = 3\hat i - 4\hat j$, find (a) $\vec A \cdot \vec B$ and (b) the angle between $\vec A$ and $\…Free
- Q20For $\vec A = 2\hat i + 3\hat j$ and $\vec B = \hat i - 2\hat j$ (both lying in the xy-plane), find $\vec A \times \vec B$, its magnitude, a…Preview
- Q21A man can swim at 4 km/h in still water; the river flows at 3 km/h. He wishes to cross the river of width 400 m by the shortest path (arrivi…Preview
- Q22Rain is falling vertically downward at 10 m/s. A man runs horizontally at 6 m/s. Find the velocity of the rain relative to the man (magnitud…Preview
- Q23A stone is thrown with a speed of 30 m/s at an angle of $45^\circ$ to the horizontal. Taking $g = 10\ \text{m/s}^2$, find the maximum height…Preview
- Q24A projectile is launched with an initial speed of 15 m/s, once at $30^\circ$ and once at $60^\circ$ to the horizontal (take $g = 10\ \text{m…Preview
- Q25A stone tied to a string of length 1.5 m is whirled in a horizontal circle at a constant speed of 6 m/s. Find the centripetal acceleration o…Preview
- Q26A point on the rim of a wheel of radius 0.4 m moves with a constant angular velocity of 5 rad/s. Find (a) its linear speed, (b) the centripe…Preview