Let -a- i - j -1k- b-b1-b1-2k- and - c -5-2k- be three vectors such that the projection vector of - b on - a is - a- If - a - b is perpendicular to - c- the - b - is equal to -

The correct option is D 6 Projection of → b on → a =→ a .→ b |→ b|=|→ a | ⇒ b_1+b_2=2 .......(1)and (→ a +→ b )→ c⇒ (→ a +→ b .→ c )= ...

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Byju's Answer

Standard XII

Mathematics

Applications of Dot Product

Let →a=→ i ...

Question

Let $\to a = \to i + \to j + \sqrt{1}^k, \to b = b_{1}^i + b_{1}^j + \sqrt{2}^k$ and $\to c = 5^i +^j + \sqrt{2}^k$ be three vectors such that the projection vector of $\to b$ on $\to a$ is $\to a$ . If $\to a + \to b$ is perpendicular to $\to c$ , the $| \to b |$ is equal to :

\sqrt{22}

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4

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\sqrt{32}

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6

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Solution

The correct option is D $6$
Projection of $\to b$ on $\to a = \frac{\to a . \to b}{| \to b |} = | \to a |$ $\Rightarrow b_{1} + b_{2} = 2$ .......(1)and $(\to a + \to b) ⊥ \to c \Rightarrow (\to a + \to b . \to c) = 0$ $\Rightarrow 5 b_{1} + b_{2} = - 10$ ......(2)from $(1)$ and $(2)$ $\Rightarrow b_{1} = - 3$ and $b_{2} = 5$ then $| \to b | = \sqrt{b_{1}^{2} + b_{2}^{2} + 2} = 6$

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Similar questions

Q. Let

\to a =^i +^j + \sqrt{2}^k, \to b = b_{1}^i + b_{2}^j + \sqrt{2}^k

and

\to c = 5^i +^j + \sqrt{2}^k

be three vectors such that the projection vector of

\to b

\to a

\to a

. If

\to a + \to b

is perpendicular to

\to c

, then

| \to b |

is equal to :

Q. Let

\to a =^i +^j + \sqrt{2}^k, \to b = b_{1}^i + b_{2}^j + \sqrt{2}^k

and

\to c = 5^i +^j + \sqrt{2}^k

be three vectors such that the projection vector of

\to b

\to a

\to a

and

(\to a + \to b)

is perpendicular to

\to c

, then

| \to b |

is equal to

Q. Let

\to a =^i +^j + \sqrt{2}^k, \to b = b_{1}^i + b_{2}^j + \sqrt{2}^k

and

\to c = 5^i +^j + \sqrt{2}^k

be three vectors such that the projection vector of

\to b

\to a

\to a

. If

\to a + \to b

is perpendicular to

\to c

, then

| \to b |

is equal to :

Q. Let

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

and

\to c =^i +^j - 2^k,

be three vectors, a vector in the plane of

b

and

c

whose projection on

\to a

is of magnitude

\sqrt{\frac{2}{3}}

Q. Let

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

and

\to b =^i +^j .

Let

\to c

be a vector such that

| \to c - \to a | = 3, | (\to a \times \to b) \times \to c | = 3

and the angle between

\to c

and

\to a \times \to b

30^{\circ} .

Then

\to a \cdot \to c

is equal to:

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Let →a=→i+→j+√1^k,→b=b1^i+b1^j+√2^k and →c=5^i+^j+√2^k be three vectors such that the projection vector of →b on →a is →a. If →a+→b is perpendicular to →c, the |→b| is equal to :

The correct option is D 6Projection of →b on →a=→a.→b|→b|=|→a|⇒b1+b2=2.......(1)and (→a+→b)⊥→c⇒(→a+→b.→c)=0⇒5b1+b2=−10......(2)from (1) and (2) ⇒b1=−3 and b2=5 then |→b|=√b21+b22+2=6

Let $\to a = \to i + \to j + \sqrt{1}^k, \to b = b_{1}^i + b_{1}^j + \sqrt{2}^k$ and $\to c = 5^i +^j + \sqrt{2}^k$ be three vectors such that the projection vector of $\to b$ on $\to a$ is $\to a$ . If $\to a + \to b$ is perpendicular to $\to c$ , the $| \to b |$ is equal to :