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Applications of Dot Product

If î + ĵ + ...

Question

If $^i +^j +^k, 2^i + 5^j, 3^i + 2^j - 3^k$ and $^i - 6^j -^k$ are the position vectors of points $A, B, C$ and $D$ respectively, then find the angle between $\to A B$ and $\to C D$

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Solution

$\to A B = i + 4 j - k$ and $\to C D = - 2 i - 8 j + 2 k$
Hence the angle between the vectors is given as
$θ = c o s^{- 1} (\frac{\to A B . \to C D}{A B . C D})$

$= c o s^{- 1} (\frac{- 2 - 32 - 2}{\sqrt{18} \times \sqrt{72}})$

$= c o s^{- 1} (\frac{- 36}{36})$

$= c o s^{- 1} (- 1)$
$= π$

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Similar questions

\to A B = 3^i - 2^j +^k

\to B C =^i + 2^j -^k

\to C D = 2^i +^j + 3^k

\to O A =^i +^j +^k

, then the position vector of

\to O D

(^i +^j +^k), (2^i + 5^j), (3^i + 2^j - 3^k)

(^i - 6^j -^k)

be the position vectors of points of points

A, B, C, D

respectively. Then show that

A B ∥ C D

If^i +^j +^k,

2^i + 5^j,

3^i + 2^j - 3^k,

^i - 6^j -^k

respectively are the position vectors of points

A,

B,

C

D,

then find the angle between the straight lines

A B

C D .

- - \to A B

- - \to C D

are collinear or not.

Q. The position vector of four points

A, B, C

D

^i +^j +^k, 2^i + 5^j, 3^i + 2^j - 3^k,^i - 6^j -^k

.
Then select the correct relation.

If^i +^j +^k,

2^i + 5^j,

3^i + 2^j - 3^k,

^i - 6^j -^k

respectively are the position vectors of points

A,

B,

C

D,

then find the angle between the straight lines

A B

C D .

- - \to A B

- - \to C D

are collinear or not.

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