For three vectors a - b and c if a - b - c and a - c - b- then prove that a - b and c are mutually perpendicular vectors- - b - a - and - a -1

We are given that .a⃗×b⃗=c⃗⇒a⃗⊥c⃗ and b⃗⊥c⃗ amp; a⃗×c⃗=c⃗⇒a⃗⊥b⃗ and c⃗⊥b⃗ amp; }⇒a⃗⊥b⃗⊥c⃗....(i) Now,a⃗×b⃗=c⃗⇒ |a⃗×b⃗|=|c⃗|⇒ |a⃗||b⃗|sin π/2=|c⃗|⇒ |a⃗||b ...

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

Byju's Answer

Standard XII

Mathematics

Applications of Dot Product

For three vec...

Question

For three vectors $\to a, \to b and \to c i f \to a \times \to b = \to c a n d \to a \times \to c = \to b$ ,then prove that $\to a, \to b a n d \to c$ are mutually perpendicular vectors, $| \to b | = | \to a | a n d | \to a | = 1$

Open in App

Solution

We are given that
$\begin{matrix} \to a \times \to b = \to c \Rightarrow \to a ⊥ \to c a n d \to b ⊥ \to c \to a \times \to c = \to c \Rightarrow \to a ⊥ \to b a n d \to c ⊥ \to b \end{matrix}} \Rightarrow \to a ⊥ \to b ⊥ \to c . . . . (i) N o w, \to a \times \to b = \to c \Rightarrow | \to a \times \to b | = | \to c | \Rightarrow | \to a | | \to b | s i n \frac{π}{2} = | \to c | \Rightarrow | \to a | | \to b | = | \to c | . . . (i i) [b y (i), \to a ⊥ \to b A n d, \to a \times \to c = \to b \Rightarrow | \to a | | \to c | s i n \frac{π}{2} = | \to b | \Rightarrow | \to a | | \to c | = | \to b | . . . (i i i) [b y (i), \to a ⊥ \to c B y (i i) \div (i i i), w e g e t : | \to c |^{2} = | \to b |^{2} \Rightarrow | \to c | = | \to b | . Substitute | \to c | = | \to b | i n (i i) to obtain, | \to a | = 1.$

Suggest Corrections

Similar questions

Q. If

\vec{a,} \vec{b,} \vec{c}

are three mutually perpendicular unit vectors, then prove that

|\vec{a} + \vec{b} + \vec{c}| = \sqrt{3}

Q. If

\vec{a}, \vec{b}, \vec{c}

are three mutually perpendicular vectors then

|\vec{a} + \vec{b} + \vec{c}|

= _______________.

Q. If

¯ ¯ ¯ a, ¯ ¯ ¯ v, ¯ ¯ c

are three mutually perpendicular vectors such that

∣ ∣ ¯ ¯ ¯ a ∣ ∣ = ∣ ∣ ¯ ¯ b ∣ ∣ = ∣ ∣ ¯ ¯ c ∣ ∣

then

(¯ ¯ ¯ a + ¯ ¯ b + ¯ ¯ c, ¯ ¯ ¯ a) =

Q. If

\to a, \to b, \to c

are three mutually perpendicular vectors of equal magnitude, prove that

(\to a + \to b + \to c)

is equally inclined with vectors

\to a, \to b

and

\to c

Q. If

\to b

and

\to c

are mutually perpendicular unit vectors and

\to a

is any vector, then

(\to a, \to b) \to b + (\to a . \to c) \to c + \frac{\to a . (\to b \times \to c)}{∣ ∣ ∣ \to b \times \to c ∣ ∣ ∣} (\to b \times \to c) =

Join BYJU'S Learning Program

For three vectors →a,→b and →c if →a×→b=→c and →a×→c=→b,then prove that →a,→b and →c are mutually perpendicular vectors,|→b|=|→a| and |→a|=1

For three vectors $\to a, \to b and \to c i f \to a \times \to b = \to c a n d \to a \times \to c = \to b$ ,then prove that $\to a, \to b a n d \to c$ are mutually perpendicular vectors, $| \to b | = | \to a | a n d | \to a | = 1$