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Q1:
Represent graphically a displacement of 40 km, 30° east of north.
Solution :
Given Variables & Initial Setup
To represent the displacement vector $\vec{d}$, we define the following parameters:
- Magnitude ($|\vec{d}|$): $40\text{ km}$. We shall use a scale factor where $1\text{ cm} = 10\text{ km}$, thus the vector length in the drawing is $4\text{ cm}$.
- Direction: $30^\circ$ East of North. This implies the angle is measured from the North (positive $y$-axis) towards the East (positive $x$-axis).
Step 1: Coordinate Geometry Calculation
To ensure high precision, we convert the polar representation $(r, \theta)$ to Cartesian coordinates $(x, y)$.
- The angle relative to the positive $x$-axis (East) is calculated as: $\theta_{std} = 90^\circ - 30^\circ = 60^\circ$. [By Complementary Angle Theorem]
- $x = r \cos(60^\circ) = 4 \times 0.5 = 2.0\text{ cm}$.
- $y = r \sin(60^\circ) = 4 \times \frac{\sqrt{3}}{2} \approx 4 \times 0.866 = 3.464\text{ cm}$.
Step 2: Graphical Representation (SVG)
Step 3: Verification of Vector Properties
The vector $\vec{d}$ is represented starting from the origin $(0,0)$ in the coordinate plane:
- The tail is fixed at the origin representing the starting point.
- The arrowhead terminates at $(2, 3.464)$ relative to the origin.
- The magnitude is validated via the Pythagorean theorem: $\sqrt{2^2 + 3.464^2} = \sqrt{4 + 12} = \sqrt{16} = 4\text{ cm}$. [By Pythagorean Identity]
Final Solution: The displacement vector is represented as a directed line segment of length 4 units, originating at the coordinate intersection and oriented at an angle of 30° East of North, equivalent to a displacement of 40 km.
More Questions from Class 12 Mathematics Vector Algebra EXERCISE 10.1
- Q2(i): Classify the following measures as scalars and vectors. (i) 10 kg
- Q2(ii): Classify the following measures as scalars and vectors. (ii) 2 meters north-west
- Q2(iii): Classify the following measures as scalars and vectors. (iii) 40°
- Q2(iv): Classify the following measures as scalars and vectors. (iv) 40 watt
- Q2(v): Classify the following measures as scalars and vectors. (v) $10^{-19}$ coulomb
- Q2(vi): Classify the following measures as scalars and vectors. (vi) $20$ m/s$^2$
- Q3(i): Classify the following as scalar and vector quantities. (i) time period
- Q3(ii): Classify the following as scalar and vector quantities. (ii) distance
- Q3(iii): Classify the following as scalar and vector quantities. (iii) force
- Q3(iv): Classify the following as scalar and vector quantities. (iv) velocity
- Q3(v): Classify the following as scalar and vector quantities. (v) work done
- Q4(i): In Fig 10.6 (a square), identify the following vectors. (i) Coinitial
- Q4(ii): In Fig 10.6 (a square), identify the following vectors. (ii) Equal
- Q4(iii): In Fig 10.6 (a square), identify the following vectors. (iii) Collinear but not equal
- Q5(i): Answer the following as true or false. (i) $\vec{a}$ and $-\vec{a}$ are collinear.
- Q5(ii): Answer the following as true or false. (ii) Two collinear vectors are always equal in magnitude.
- Q5(iii): Answer the following as true or false. (iii) Two vectors having same magnitude are collinear.
- Q5(iv): Answer the following as true or false. (iv) Two collinear vectors having the same magnitude are equal.
CBSE Solutions for Class 12 Mathematics Vector Algebra
Chapters in CBSE - Class 12 Mathematics
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