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Addition of Vectors

If a⃗ and ...

Question

If $\to a$ and $\to b$ are the vectors determined by two adjacent sides of regular hexagon, then vector $E F$ is

A

(\to a + \to b)

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B

(\to a - \to b)

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C

2 \to a

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D

2 \to b

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Solution

The correct option is B $(\to a - \to b)$
FABCDE is a regular hexgon.

Let $¯ ¯¯¯¯¯¯ ¯ F A = \to a$ and $¯ ¯¯¯¯¯¯ ¯ A B = \to b$ , join $F B$ and $F C$ , we have
$¯ ¯¯¯¯¯¯ ¯ F B = ¯ ¯¯¯¯¯¯ ¯ F A + ¯ ¯¯¯¯¯¯ ¯ A B = \to a + \to b$

Since, $¯ ¯¯¯¯¯¯ ¯ F C$ is parallel to $¯ ¯¯¯¯¯¯ ¯ A B$ and double of

$¯ ¯¯¯¯¯¯ ¯ A B$
$∴ ¯ ¯¯¯¯¯¯ ¯ F C = 2 ¯ ¯¯¯¯¯¯ ¯ A B = 2 \to b$

Now, $¯ ¯¯¯¯¯¯ ¯ B C = F C - F B$ $= 2 \to b - (\to a + \to b) = \to b - \to a$

$¯ ¯¯¯¯¯¯¯ ¯ C D = - ¯ ¯¯¯¯¯¯ ¯ F A = - \to a$ and $¯ ¯¯¯¯¯¯¯ ¯ D E = - ¯ ¯¯¯¯¯¯ ¯ A B = - \to b$

Also, $¯ ¯¯¯¯¯¯ ¯ E F = - ¯ ¯¯¯¯¯¯ ¯ B C = - (\to b - \to a) = \to a - \to b$

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Similar questions

Q. Match List 1 with List 2

	List 1		List 2
A.	The volume of the parallelopiped determined by vectors $\to a, \to b$ and $\to c$ is 2. Then, the volume of the parallelopiped determined by vectors $2 (\to a \times \to b), 3 (\to b \times \to c)$ and $(\to c \times \to a)$ is	1.	$100$
B.	The volume of the parallelopiped determined by vectors $\to a, \to b$ and $\to c$ is 5. Then, the volume of the parallelopiped determined by vectors $3 (\to a + \to b), (\to b + \to c)$ and $2 (\to c + \to a)$ is	2.	$30$
C.	The area of a triangle with adjacent sides determined by vectors $\to a$ and $\to b$ is 20. Then, the area of the triangle with adjacent sides determined by vectors $(2 \to a + 3 \to b)$ and $(\to a - \to b)$ is	3.	$24$
D.	The area of a parallelogram with adjacent sides determined by vectors $\to a$ and $\to b$ is $30$ . Then, the area of the parallelogram with adjacent sides determined by vectors $(\to a + \to b)$ and $\to a$ is	4.	$60$

\to a, \to b

| \to a | = \sqrt{3}, | \to b | = 2

(\to a, \to b) = \frac{π}{3}

. Then the area of the triangle with adjacent sides

\to a + 2 \to b

2 \to a + \to b

(\to a, \to b) = \frac{π}{3}

2 \to a + \to b, \to a - 2 \to b

are the adjacent sides of a parallelogram and if

| \to a | = | \to b | = 1

, then the lengths of the diagonals are

\to a

\to b

are unit vectors such that

(\to a + \to b) . (2 \to a + 3 \to b) \times (3 \to a - 2 \to b) = 0

\to a

\to b

be two vectors such that

| \to a | = 3

| \to b | = 6

| \to a + \to b | = 7

. Then the value of

(3 \to a - 2 \to b) . (2 \to a + 5 \to b - 4 \to a \times \to b)

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