Find a vector r- in the plane of p- and q-k- such that r- is perpendicular to p- and r-q-2 -

p⃗ = ̂1̂+ĵ q⃗ = ̂ĵ+k̂ Let r⃗ = xî+yĵ+zk̂ Given p⃗⊥ #160;r⃗⇒p̂.r̂ = 0⃗ ⇒ x+y = 0 #160; ⇒ x = y and r⃗.q⃗ = 2 ⇒ y ...

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Byju's Answer

Standard XII

Mathematics

Perpendicular Distance of a Point from a Plane

Find a vector...

Question

Find a vector $\to r$ in the plane of $\to p = -^i +^j$ and $\to q = -^j +^k$ such that $\to r$ is perpendicular to $\to p$ and $\to r . \to q = - 2$ .

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Solution

$\to p =^- 1 +^j$
$\to q =^- j +^k$
Let $\to r = x^i + y^j + z^k$
Given $\to p ⊥ \to r \Rightarrow^p .^r = \to 0$
$\Rightarrow - x + y = 0$
$\Rightarrow x = y$
and $\to r . \to q = - 2$
$\Rightarrow - y + z = - 2$
$\Rightarrow z = y - 2$
$\Rightarrow x = y$ and $z = y - 2$
Let $x = 1 \Rightarrow y = 1$ & $z = - 1$
vector $\to r =^i +^j -^k$

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Q. Find a vector

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\to p = -^i +^j

and

\to q = -^j +^k

such that

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∣ ∣ ∣ ∣ \begin{matrix} \to p . \to p & \to p . \to q & \to p . \to r \to p . \to q & \to q . \to q & \to q . \to r \to p . \to r & \to r . \to q & \to r . \to r \end{matrix} ∣ ∣ ∣ ∣

Q. If

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Find a vector →r in the plane of →p=−^i+^j and →q=−^j+^k such that →r is perpendicular to →p and →r.→q=−2 .

→p=^−1+^j→q=^−j+^kLet →r=x^i+y^j+z^kGiven →p⊥→r⇒^p.^r=→0⇒−x+y=0 ⇒x=yand →r.→q=−2⇒−y+z=−2⇒z=y−2 ⇒x=y and z=y−2Let x=1⇒y=1 & z=−1 vector →r=^i+^j−^k

Find a vector $\to r$ in the plane of $\to p = -^i +^j$ and $\to q = -^j +^k$ such that $\to r$ is perpendicular to $\to p$ and $\to r . \to q = - 2$ .