Let a- - i - 2j - k- b- - i - j - k and c- - i - j - k- a vector in the plane a- and b- whose projection on c- is 1-3 is

The correct option is C 2i + j + 2k A vector in the plane of a⃗ and b⃗ would be ma⃗ + nb⃗ i.e. (m + n)i + (2m n)j + (m + n)k Its p ...

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Byju's Answer

Standard XII

Mathematics

Applications of Dot Product

Let a⃗ = i ...

Question

Let $\to a = i + 2 j + k, \to b = i - j + k$ and $\to c = i + j - k$ , a vector in the plane $\to a$ and $\to b$ whose projection on $\to c$ is $\frac{1}{\sqrt{3}}$ is _____

3 i + j - 3 k

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4 i + j - 4 k

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i + j - 2 k

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2 i + j + 2 k

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Solution

The correct option is C $2 i + j + 2 k$
A vector in the plane of $\to a$ and $\to b$ would be $m \to a + n \to b$
i.e. $(m + n) i + (2 m - n) j + (m + n) k$
Its projection on $\to c$ will be $\frac{(m \to a + n \to b) . \to c}{| c |}$
$∴ \frac{1}{\sqrt{3}} = \frac{[(m + n) i + (2 m - n) j + (m + n) k] . (i + j - k)}{\sqrt{3}}$
$\Rightarrow 1 = m + n + 2 m - n - m - n$
$\Rightarrow 1 = 2 m - n$
This relation is satisfied by $m = 1, n = 1$
Thus, the required vector can be $2 i + j + 2 k$

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Let →a=i+2j+k,→b=i−j+k and →c=i+j−k, a vector in the plane →a and →b whose projection on →c is 1√3 is _____

Let $\to a = i + 2 j + k, \to b = i - j + k$ and $\to c = i + j - k$ , a vector in the plane $\to a$ and $\to b$ whose projection on $\to c$ is $\frac{1}{\sqrt{3}}$ is _____