Q5. In triangle ABC, D is the midpoint of BC. Show that AB + AC = 2AD.
AB = AD + DB and AC = AD + DC. Since D is the midpoint, DC = −DB, so the sum is 2AD. (It is also the parallelogram law: AB + AC is the diagonal, twice the median.)
Q6. Explain, with a parallelogram, why a + b = b + a (NCERT Property 1, page 344).
Draw ABCD with AB = a and BC = b. Opposite sides give AD = b and DC = a. The triangle law in ABC gives AC = a + b, and in ADC gives AC = b + a, so the two are equal.
Q7. When is |a + b| = |a| + |b|? When is |a + b| = |a| − |b|, with |a| ≥ |b|?
The first when a and b have the same direction (or one is 0): the triangle collapses to a straight line. The second when they have opposite directions, or when b = 0 (including both zero): then the shorter one is taken off the longer.
Q8. A boat can move at 13 km/h in still water. The river flows at 5 km/h. In what direction should it head to cross straight over, and at what speed does it then cross?
Its own velocity must cancel the flow along the river: if it heads at angle θ upstream of straight across, 13 sin θ = 5, so θ ≈ 22.6°. The crossing speed is 13 cos θ = 12 km/h.
Q9. Show that a − b and b − a are negatives of each other, and find |b − a| if |a − b| = 6.
(a − b) + (b − a) = 0, so each is the negative of the other. Negatives have equal magnitudes, so |b − a| = 6.
Q10. Beyond the NCERT text (JEE): if |a + b| = |a − b|, what can you say about the parallelogram on a and b?
If both vectors are nonzero, its equal diagonals make it a rectangle and a, b are perpendicular. If either vector is 0, the equality still holds but the figure is degenerate and no direction is assigned to that zero vector. Example 3 and practice Q2 are such cases.