Let x- y and z be three vectors each of magnitude -2 and the angle between each pair of them is -3- If a is a non-zero vector perpendicular to x and y-z and b is a non-zero vector perpendicular to y and z-x- then

The correct options are A b = (b·z) (z x) B a = (a·y) (y z) C a·b = (a·y) (b·z) a is in direction of x× (y × #160;z) i.e. ...

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Byju's Answer

Standard XII

Mathematics

Equation of a Plane : Vector Form

Let x, y an...

Question

Let $\to x, \to y$ and $\to z$ be three vectors each of magnitude $\sqrt{2}$ and the angle between each pair of them is $\frac{π}{3}$ . If $\to a$ is a non-zero vector perpendicular to $\to x$ and $\to y \times \to z$ and $\to b$ is a non-zero vector perpendicular to $\to y$ and $\to z \times \to x$ , then

\to b = (\to b \cdot \to z) (\to z - \to x)

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\to a = (\to a \cdot \to y) (\to y - \to z)

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\to a \cdot \to b = - (\to a \cdot \to y) (\to b \cdot \to z)

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\to a = - (\to a \cdot \to y) (\to z - \to y)

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Solution

The correct options are
A $\to b = (\to b \cdot \to z) (\to z - \to x)$
B $\to a = (\to a \cdot \to y) (\to y - \to z)$
C $\to a \cdot \to b = - (\to a \cdot \to y) (\to b \cdot \to z)$
$\to a$ is in direction of $\to x \times (\to y \times \to z)$
i.e.
$(\to x \cdot \to z) \to y - (\to x \cdot \to y) \to z$
$\Rightarrow \to a = λ_{1} [2 \times \frac{1}{2} (\to y - \to z)]$
$\to a = λ_{1} (\to y - \to z)$ ...... $(1)$
Now
$\to a \cdot \to y = λ_{1} (\to y \cdot \to y - \to y \cdot \to z)$
$= λ_{1} (2 - 1) \Rightarrow λ_{1} = \to a \cdot \to y$ ...... $(2)$
From $(1)$ and $(2)$ ,
$\to a = \to a \cdot \to y (\to y - \to z)$
Similarly, $\to b = (\to b \cdot \to z) (\to z - \to x)$
Now,
$\to a \cdot \to b = (\to a \cdot \to z) [(\to y - \to z) \cdot (\to z - \to x)]$
$= (\to a \cdot \to y) (\to b \cdot \to z) [1 - 1 - 2 + 1]$
$= - (\to a \cdot \to y) (\to b \cdot \to z)$ .
$\to a$

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Similar questions

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\to x, \to y

and

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and the angle between each pair of them is

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\to y \times \to z

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Let →x,→y and →z be three vectors each of magnitude √2 and the angle between each pair of them is π3. If →a is a non-zero vector perpendicular to →x and →y×→z and →b is a non-zero vector perpendicular to →y and →z×→x, then