Let a-b-c- be vectors of length 3- 4- 5 respectively- Let a- be perpendicular to b-c- b- to c-a- c- to a-b- Then - a-b-c- - is-

The correct option is D 5√(2) a⃗·( b⃗ +c⃗ #160;) = 0 or a⃗ .b⃗ +a⃗ .c⃗ = 0 b⃗·( a⃗ +c⃗ #160;) = 0 or a⃗ .b⃗ +b⃗ .c⃗ = 0 c⃗·( a⃗ +b⃗ ...

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Byju's Answer

Standard XII

Mathematics

Integration of Irrational Algebraic Fractions - 2

Let a⃗,b⃗,c...

Question

Let $\to a, \to b, \to c$ be vectors of length $3, 4, 5$ respectively. Let $\to a$ be perpendicular to $\to b + \to c, \to b$ to $\to c + \to a$ & $\to c$ to $\to a + \to b .$ Then $∣ ∣ \to a + \to b + \to c ∣ ∣$ is:

2 \sqrt{5}

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2 \sqrt{2}

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10 \sqrt{5}

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5 \sqrt{2}

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Solution

The correct option is D $5 \sqrt{2}$
$\to a \cdot (\to b + \to c) = 0 o r \to a . \to b + \to a . \to c = 0 \to b \cdot (\to a + \to c) = 0 o r \to a . \to b + \to b . \to c = 0 \to c \cdot (\to a + \to b) = 0 o r \to a . \to c + \to b . \to c = 0$
Add the three to get $(\to a . \to b + \to b . \to c + \to a . \to c) = 0$
Now, ${(\to a + \to b + \to c)}^{2} = a^{2} + b^{2} + c^{2} + 2 (\to a . \to b + \to b . \to c + \to a . \to c) = a^{2} + b^{2} + c^{2} = 9 + 16 + 25 = 50$
Thus, $| {(\to a + \to b + \to c)}^{2} | = 50 o r | (\to a + \to b + \to c) | = \sqrt{50} = 5 \sqrt{2}$

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Similar questions

Q. For any three vectors

\to a, \to b

and

\to c

, prove that

[\to a + \to b, \to b + \to c, \to c + \to a] = 2 [\to a \to b \to c]

. Hence prove that the vectors

\to a + \to b, \to b + \to c, \to c + \to a

are coplanar. If and only if

\to a, \to b, \to c

are coplanar.

Q. If

\to a, \to b, \to c

are vectors of lengths

2, 3, 4

. If

\to a

is orthogonal to

\to b + \to c, \to b

is orthogonal to

\to c + \to a

and

\to c

is orthogonal to

\to a + \to b

, then the modulus of

\to a + \to b + \to c

Q. Let

\to a, \to b, \to c

be vectors of length

3, 4

and

5

respectively. Let

\to a

be perpendicular to

(\to b + \to c), \to b

(\to c + \to a)

and

\to c

(\to a + \to b)

. Find the length of the vector

\to a + \to b + \to c

Q. Let

\to a, \to b

and

\to c

be three vectors such that

| \to a | = \sqrt{3}, | \to b | = 5, \to b . \to c = 10

and the angle between

\to b

and

\to c

\frac{π}{3} .

\to a

is perpendicular to vector

\to b \times \to c

, then

| \to a \times (\to b \times \to c) |

is equal to

Q. If

| \to a | = 3, ∣ ∣ \to b ∣ ∣ = 4, | \to c | = 5, \to a ⊥ (\to b + \to c), \to b ⊥ (\to c + \to a)

and

\to c ⊥ (\to a + \to b)

then

\sqrt{2} ∣ ∣ \to a + \to b + \to c ∣ ∣

is equal to

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Let →a,→b,→c be vectors of length 3,4,5 respectively. Let →a be perpendicular to →b+→c,→b to →c+→a & →c to →a+→b. Then ∣∣→a+→b+→c∣∣ is:

Let $\to a, \to b, \to c$ be vectors of length $3, 4, 5$ respectively. Let $\to a$ be perpendicular to $\to b + \to c, \to b$ to $\to c + \to a$ & $\to c$ to $\to a + \to b .$ Then $∣ ∣ \to a + \to b + \to c ∣ ∣$ is: