Q.Represent graphically a displacement of 40 km, $30^\circ$ east of north.

Concept: Vector Component Extraction – A vector is resolved into perpendicular components to specify its exact direction and magnitude.

Reasoning:

1. The direction “$30^\circ$ east of north” means the vector makes a $30^\circ$ angle with the north axis, turning toward east.
2. The north component is the adjacent side: $40 \cos 30^\circ = 40 \times \frac{\sqrt{3}}{2} = 20\sqrt{3}$ km.
3. The east component is the opposite side: $40 \sin 30^\circ = 40 \times \frac{1}{2} = 20$ km.

Graphical representation: Draw a vertical arrow (north) of length $20\sqrt{3}$ units, then from its tip a horizontal arrow (east) of length $20$ units. The resultant vector from the origin to the end of the east arrow is the required displacement.

The key idea is that a vector has both magnitude and direction. To represent it graphically, we need a scale and a clear reference for the angle.

Why this approach works:

A displacement vector tells you how far and in what direction something moves. Drawing it as an arrow makes the magnitude visible through length and the direction through the angle the arrow makes with a chosen axis. The phrase "$30^\circ$ east of north" means you start facing north, then turn $30^\circ$ toward the east. So the arrow points northeast-ish, but closer to north than to east.

Let’s build the representation step by step.

1. Choose a scale.

   The magnitude is 40 km. We need a convenient scale so the arrow fits on paper.

   Let 1 cm = 10 km. Then 40 km becomes 4 cm.

   Tip

   Always pick a scale that makes the arrow large enough to draw accurately but small enough to fit. 1 cm = 10 km is clean here.

2. Draw the reference axes.

   Mark a point as the origin (starting point). Draw a vertical line upward for north and a horizontal line to the right for east. Label them clearly.

3. Interpret the direction.

   "30° east of north" means:

   • Start from the north direction (pointing straight up).
   • Rotate 30° toward the east (clockwise from north). So the arrow lies in the first quadrant (north-east quadrant), making a 30° angle with the north axis.
   Watch out

   A common mistake is to measure the angle from the east axis instead. "East of north" always means the angle is measured from north toward east. If it said "north of east," you would measure from east toward north.

4. Draw the arrow.

   From the origin, draw a straight line of length 4 cm at an angle of 30° measured from the north direction toward the east.

   • Use a protractor: align its center at the origin, put the 0° mark on the north line, then mark 30° toward the east.
   • Draw the line along that mark, 4 cm long.
   • Add an arrowhead at the tip to show the direction of displacement.

5. Label the vector.

   Write the magnitude (40 km) and the angle (30°) near the arrow. You can also write the vector name, say $\vec{d}$, above the arrow.

The final diagram looks like this (described in words):

        North
          |
          |  / 
          | /   (arrow at 30° from north toward east)
          |/ 30°
   Origin *-------> East

The arrow is 4 cm long, pointing 30° east of north.

Method: Drawing a displacement vector to scale from a compass bearing

Use this for any "represent graphically a displacement of $D$ at a bearing" question — the goal is an accurate scaled arrow, not a calculation.

Steps

Step 1: Choose a scale and convert the magnitude to a drawable length.

Pick a clean scale (e.g. $1\text{ cm}=10\text{ km}$) so the arrow is neither tiny nor off the page, then convert: $40\text{ km}\to 4\text{ cm}$.

Step 2: Set up the reference directions and decode the bearing.

Draw north pointing up and east pointing right from a chosen origin. Read the bearing carefully: "$\theta$ east of north" means start along the north line and rotate $\theta$ toward the east. (Contrast: "north of east" would be measured from the east line instead.)

Step 3: Lay off the angle and draw the arrow.

With a protractor centred at the origin and $0^\circ$ on the correct reference line, mark the angle, draw the segment to the scaled length, and add an arrowhead at the tip. Label both the magnitude and the angle so the diagram is self-describing.

Common Mistakes

Mistake 1: Measuring the $30^\circ$ from the east axis instead of from north.

Why it's wrong: "east of north" means the angle is taken from the north line, rotating toward east; measuring from east gives a completely different arrow. Correct approach: start the protractor's $0^\circ$ on north and turn $30^\circ$ toward east.

Mistake 2: Drawing the arrow without a stated scale.

Why it's wrong: a graphical representation must show magnitude through length; an arbitrary length conveys no information about $40$ km. Correct approach: declare a scale (e.g. $1\text{ cm}=10\text{ km}$) and draw the length accordingly.

Mistake 3: Omitting the arrowhead or labels.

Why it's wrong: without an arrowhead the direction is ambiguous, and without the magnitude/angle labels the diagram is incomplete. Correct approach: add the arrowhead at the tip and annotate both $40$ km and $30^\circ$.