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Applications of Dot Product

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Question

Show that the points with position vectors $-^i + 6^j + 6^k, - 4^i + 9^j + 6^k,$ and $7^j + 10^k$ from a right angled triangle.

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Solution

$\begin{matrix} A (- ˆ i + 6 ˆ j + 6 ˆ k) B (- 4 ˆ i + 9 ˆ j + 6 ˆ k) C (7 ˆ j + 10 ˆ k) - - \to A B = - 3 ˆ i + 3 ˆ j - - \to A C = ˆ i + ˆ j + 4 ˆ k - - \to A B . - - \to A C = 0 T h u s A, B a n d C f o r a n r i g h t a n g l e d t r i a n g l e . \end{matrix}$

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A B C

be a triangle, the position vector of whose vertices are respectively

- 10^i + 10^k, -^i + 6^j + 6^k

- 4^i + 9^j + 6^k

△ A B C

Q. Show that the four points whose position vectors are

6^i - 7^j, 16^i - 29^j - 4^k, 3^j - 6^k

2^i + 5^j + 10^k

Q. The position vectors of points

A, B, C

D

A = 3^i + 4^j + 5^k

B = 4^i + 5^j + 6^k, C = 7^i + 9^j + 3^k

D = 4^i + 6^j

, then the displacement vectors

- - \to A B

- - \to C D

Q. The position vectors of points

A, B, C

D

A = 3^i + 4^j + 5^k, B = 4^i + 5^j + 6^k

C = 7^i + 9^j + 3^k

D = 4^i + 6^j

. Then, the displacement vectors

A B

C D

Q. The position vectors of points

A, B, C

D

A = 3^i + 4^j + 5^k, B = 4^i + 5^j + 6^k

C = 7^i + 9^j + 3^k

D = 4^i + 6^j

. Then, the displacement vectors

A B

C D

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