Let three vectors a-b and c be such that c is coplanar with a and b- a- c-7 and b is perpendicular to c- where a-k- and b-2-k- then the value of 2-a-b-c-2 is

Let c=λ(b×(a×b)) =λ((b.b)a (b.a)b) =5λ(( î+ĵ+k̂)+2î+k̂) =λ( 3î+5ĵ+6k̂) c.a=7⇒ 3λ+5λ+6λ=7 λ=12 ∴ 2 | ( 32 1+2 )î+ ( 52+1 )ĵ+(3+1+1)k̂ nbsp; |^2 =2 ( 14+494 ...

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Byju's Answer

Standard XII

Mathematics

Angle between Two Vectors

Let three vec...

Question

Let three vectors $\to a, \to b$ and $\to c$ be such that $\to c$ is coplanar with $\to a$ and $\to b, \to a . \to c = 7$ and $\to b$ is perpendicular to $\to c,$ where $\to a = -^i +^j +^k$ and $\to b = 2^i +^k$ , then the value of $2 | \to a + \to b + \to c |^{2}$ is

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Solution

Let $\to c = λ (\to b \times (\to a \times \to b))$
$= λ ((\to b . \to b) \to a - (\to b . \to a) \to b)$
$= 5 λ ((-^i +^j +^k) + 2^i +^k)$
$= λ (- 3^i + 5^j + 6^k)$
$\to c . \to a = 7 \Rightarrow 3 λ + 5 λ + 6 λ = 7$
$λ = \frac{1}{2}$
$∴ 2 {∣ ∣ ∣ (- \frac{3}{2} - 1 + 2)^i + (\frac{5}{2} + 1)^j + (3 + 1 + 1)^k ∣ ∣ ∣}^{2}$
$= 2 (\frac{1}{4} + \frac{49}{4} + 25) = 25 + 50 = 75$

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Similar questions

Q. Let three vectors

\to a, \to b

and

\to c

be such that

\to c

is coplanar with

\to a

and

\to b, \to a . \to c = 7

and

\to b

is perpendicular to

\to c,

where

\to a = -^i +^j +^k

and

\to b = 2^i +^k

, then the value of

2 | \to a + \to b + \to c |^{2}

a = -^i +^j +^k

and

b = 2^i +^j +^k

, then find the vector

c

satisfying the conditions

(i)

that it is coplanar with

a

and

b

(i i)

that it is perpendicular to

b

(i i i

)

a . c = 7

Q. Let a vector

\to a

be coplanar with vectors

\to b = 2^i +^j +^k

and

\to c =^i -^j +^k .

\to a

is perpendicular to

\to d = 3^i + 2^j + 6^k

, and

| \to a | = \sqrt{10} .

Then a possible value of

[\to a \to b \to c] + [\to a \to b \to d] + [\to a \to c \to d]

is equal to

Q. Let

\to c

be a unit vector which is coplanar with

\to a =^i -^j + 2^k

and

\to b = 2^i -^j +^k

such that

\to c

is perpendicular to

\to a

. If

P

be the length projection of

\to c

along

\to b

, then the value of

\frac{11}{P^{2}}

Q. If

\to a =^i +^j -^k, \to b =^i -^j +^k

, and

\to c

is unit vector perpendicular to the vector

\to a

and coplanar with

\to a

and

\to b

then a unit vector

\to d

perpendicular to both

\to a

and

\to c

, is

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Let three vectors →a,→b and →c be such that →c is coplanar with →a and →b, →a.→c=7 and →b is perpendicular to →c, where →a=−^i+^j+^k and →b=2^i+^k, then the value of 2|→a+→b+→c|2 is

Let →c=λ(→b×(→a×→b)) =λ((→b.→b)→a−(→b.→a)→b) =5λ((−^i+^j+^k)+2^i+^k) =λ(−3^i+5^j+6^k) →c.→a=7⇒3λ+5λ+6λ=7 λ=12 ∴2∣∣∣(−32−1+2)^i+(52+1)^j+(3+1+1)^k∣∣∣2 =2(14+494+25)=25+50=75

Let three vectors $\to a, \to b$ and $\to c$ be such that $\to c$ is coplanar with $\to a$ and $\to b, \to a . \to c = 7$ and $\to b$ is perpendicular to $\to c,$ where $\to a = -^i +^j +^k$ and $\to b = 2^i +^k$ , then the value of $2 | \to a + \to b + \to c |^{2}$ is