Which way should you tilt your umbrella when the wind blows?
Tilt it towards the direction the wind is coming from. In NCERT Example 3.1, rain falls at 35 m s⁻¹ and a wind blows at 12 m s⁻¹ from east to west. Adding the two velocities as vectors gives a resultant of 37 m s⁻¹ at about 19° to the vertical, slanting towards the west. So the umbrella should lean about 19° from the vertical with its top towards the east.
Why you can't just add 35 and 12
Add the numbers and you get 47 m s⁻¹, which is wrong. The rain's velocity points straight down and the wind's points sideways. Two velocities at right angles don't pile up into one bigger number; they combine into a slanted one. Section 3.4 shows how.
Method one: head to tail
Draw the first vector. Put the tail of the second at the head of the first. The resultant runs from the free tail to the free head, closing the triangle. This is the triangle law; NCERT calls it the head-to-tail method.
For the rain: draw 35 units straight down, then 12 units pointing west from its tip. Join start to finish. That slanted line is the velocity the drops actually have.
Method two: the parallelogram
Put both vectors tail to tail and complete the parallelogram. The diagonal from the common tail is the resultant. Both methods always agree, so use whichever is easier to draw.
The numbers in Example 3.1
The two velocities are perpendicular, so Pythagoras does the job:
R = √(35² + 12²) = √(1225 + 144) = √1369 = 37 m s⁻¹
It comes out exact, which is rare enough to enjoy.
For the direction, measure the angle θ from the vertical:
tan θ = 12 / 35 = 0.343, so θ ≈ 19°
The rain hits you slanting towards the west, so you meet it by pointing the top of the umbrella the other way, towards the east. Point it west and the drops come in under the edge.
Note that the book solves this purely by adding two velocities. It doesn't frame it as a relative-velocity problem.
Rules vector addition follows
- Order doesn't matter: A + B = B + A (commutative)
- Grouping doesn't matter: (A + B) + C = A + (B + C) (associative)
Add a vector to its own opposite, A + (−A), and you get a vector of magnitude zero. NCERT calls it the null vector and notes that its direction cannot be specified. An arrow of zero length points nowhere in particular.
Subtraction works through addition: A − B means A + (−B). Flip B round and add head to tail.
One common trap: the magnitude of A + B is not |A| + |B| in general. That simple sum works only when the vectors point the same way. At any other angle the resultant is shorter, as the rain shows: 37 is well short of 47.
Check yourself
You walk 3 m east, then 4 m north. What is your displacement? And if A is 3 m east and B is 4 m north, what is the size of A − B? (Answers at the bottom.)
Questions students ask
Which way should an umbrella be tilted in rain with wind?
Towards the side the wind blows from. In NCERT Example 3.1 the wind blows east to west, so the umbrella tilts about 19° from the vertical towards the east.
What is the triangle law of vector addition?
Place the tail of the second vector at the head of the first. The resultant goes from the tail of the first to the head of the second.
Do the triangle and parallelogram methods give the same answer?
Yes. They are two drawings of the same rule.
What is a null vector?
A vector with zero magnitude, such as A + (−A). Its direction cannot be specified.
Is |A + B| equal to |A| + |B|?
Only if the two vectors point in the same direction. Otherwise it is smaller.
Answers: √(3² + 4²) = 5 m, at about 53° north of east. A − B is 3 m east plus 4 m south, so its size is also 5 m.
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