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If in a right-angled triangle `A B C ,` the hypotenuse `A B=p ,t h e n vec A BdotA C+ vec B Cdot vec B A+ vec C Adot vec C B` is equal to `2p^2` b. `(p^2)/2` c. `p^2` d. none of these

A

`2p^(2)`

B

`p^(2)/2`

C

`p^(2)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
c
we have
` vec(AB). Vec(AC) + vec(BC) + vec(BA) + vec(CA) . Vec(CB)`
` AB ( AC cos theta + BC sin theta)`
` AB ( AC cos theta + BC sin theta)`
`AB (((AC)^(2))/(AB)) + ((BC)^(2))/(AB)`
` AC^(2) + BC^(2) = AB^(2) + p^(2)`
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CENGAGE-DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS -Exercise
  1. Given three vectors eveca, vecb and vecc two of which are non-collinea...

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  2. If veca and vecb are unit vectors such that (veca +vecb). (2veca + 3ve...

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  3. If in a right-angled triangle A B C , the hypotenuse A B=p ,t h e n...

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  4. Resolved part of vector veca and along vector vecb " is " veca1 and th...

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  5. Let veca=2hati=hatj+hatk, vecb=hati+2hatj-hatk and vecc=hati+hatj-2hat...

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  6. If P is any arbitrary point on the circumcirlce of the equllateral ...

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  7. If vec ra n d vec s are non-zero constant vectors and the scalar b...

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  8. veca and vecb are two unit vectors that are mutually perpendicular. A...

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  9. Given that veca,vecb,vecp,vecq are four vectors such that veca + vecb...

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  10. The position vectors of the vertices A ,Ba n dC of a triangle are t...

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  11. If a is real constant A ,Ba n dC are variable angles and sqrt(a^2-4)ta...

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  12. The vertex A triangle A B C is on the line vec r= hat i+ hat j+lambda...

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  13. A non-zero vector vec a is such that its projections along vectors...

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  14. Position vector hat k is rotated about the origin by angle 135^0 i...

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  15. In a quadrilateral A B C D , vec A C is the bisector of vec A Ba n d ...

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  16. In fig. 2.33 AB, DE and GF are parallel to each other and AD, BG and E...

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  17. A unit vector veca in the plane of vecb=2hati+hatj and vecc=hati-hatj+...

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  18. Let A B C D be a tetrahedron such that the edges A B ,A Ca n dA D a...

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  19. Let vecf(t)=[t] hat i+(t-[t]) hat j+[t+1] hat k , w h e r e[dot] deno...

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  20. If veca is parallel to vecb xx vecc, then (veca xx vecb) .(veca xx vec...

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