CameraIcon

SearchIcon

MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Position Vector

Let x, y and ...

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 nonzero vector perpendicular to $\to x$ and $\to y \times \to z$ and $\to b$ is a nonzero vector perpendicular to $\to y$ and $\to z \times \to x$ , then

A

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

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

B

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

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

C

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

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

D

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

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is C $\to a \cdot \to b = - (\to a \cdot \to y) (\to b \cdot \to z)$
$| \to x | = | \to y | = | \to z | = \sqrt{2} \to x \cdot \to y = \to y \cdot \to z = \to x \cdot \to z = \sqrt{2} \times \sqrt{2} \times \frac{1}{2} = 1$
Let,
$\to a = α (\to x \times (\to y \times \to z))$
Taking dot product with $\to y$ ,
$\to a \cdot \to y = α \to y \cdot [(\to x \cdot \to z) \to y - (\to x \cdot \to y) \to z] \Rightarrow \to a \cdot \to y = α \to y \cdot [\to y - \to z] \Rightarrow \to a \cdot \to y = α [(\sqrt{2})^{2} - 1] \Rightarrow \to a \cdot \to y = α$
$∴ \to a = (\to a \cdot \to y) (\to y - \to z)$

Similarly let
$\to b = β (\to y \times (\to z \times \to x))$
Taking dot product with $\to z$ ,
$β = \to b \cdot \to z$
$∴ \to b = (\to b \cdot \to z) (\to z - \to x)$

So,
$\to a \cdot \to b = (\to a \cdot \to y) (\to b \cdot \to z) [(\to y - \to z) \cdot (\to z - \to x)] \Rightarrow \to a \cdot \to b = - (\to a \cdot \to y) (\to b \cdot \to z)$

Suggest Corrections

thumbs-up

10

Similar questions

\to x, \to y

\to z

be three vectors each of magnitude

\sqrt{2}

and the angle between each pair of them is

\frac{π}{3}

\to a

is a non-zero vector perpendicular to

\to x

\to y \times \to z

\to b

is a non-zero vector perpendicular to

\to y

\to z \times \to x

\to x, \to y

\to z

be unit vectors such that

\to x + \to y + \to z = \to a, \to x \times (\to y \times \to z) = \to b, (\to x \times \to y) \times \to z = \to c

\to a \cdot \to x = \frac{3}{2}, \to a \cdot \to y = \frac{7}{4}

| \to a | = 2

.

Then which of the following option(s) is/are CORRECT ?

Q. Let the unit vectors

\to a

\to b

be perpendicular to eachs other and the unit vector

\to c

be inclined at an at an angle

θ

\to a

\to b

x \to a + y \to b + z (\to a \times \to b)

\to a, \to b, \to c

be non-coplanar vectors and

\to d

be a non-zero vector which is perpendicular to

\to a + \to b + \to c

\to d = x \to b \times \to c + y \to c \times \to a + z \to a \times \to b

\to x \times \to y = \to a, \to y \times \to z = \to b, \to x \cdot \to b = γ, \to x \cdot \to y = 1

\to y \cdot \to z = 1

\to z

View More

Join BYJU'S Learning Program

explore_more_icon

Explore more

Position Vector

Standard XII Physics

Join BYJU'S Learning Program

CrossIcon