First slide

Algebra of vectors

Question

ABCD a parallelogram, and A₁ and B₁ are the midpoints of sides BC and CD, respectively If ${\vec{AA}}_{1} + {\vec{AB}}_{1} = λ \vec{AC}, then λ$ is equal to

Moderate

A

$\frac{1}{2}$
B

1
C

$\frac{3}{2}$
D

2

Solution

Let P.V. of A, B and D be $\vec{0}, \vec{b} and \vec{d}$ respectively.
Then P.V. of C, $\vec{c} = \vec{b} + \vec{d}$
Also P.V of $A_{1} = \vec{b} + \frac{\vec{d}}{2}$
and P.V. of $B_{1} = \vec{d} + \frac{\vec{b}}{2}$
$\Rightarrow {\vec{AA}}_{1} + {\vec{AB}}_{1} = \frac{3}{2} (\vec{b} + \vec{d}) = \frac{3}{2} \vec{AC}$

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