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O ɪs ᴛʜᴇ ᴏʀɪɢɪɴ ᴀɴᴅ A ɪs (ᴀ,ʙ,ᴄ). Fɪɴᴅ ᴛʜᴇ ᴅɪʀᴇᴄᴛɪᴏɴ ᴄᴏsɪɴᴇs ᴏғ ᴛʜᴇ ʟɪɴᴇ OA ᴀɴᴅ ᴛʜᴇ ᴇϙᴜᴀᴛɪᴏɴ ᴏғ ᴛʜᴇ ᴘʟᴀɴᴇ ᴛʜʀᴏᴜɢʜ A ᴀᴛ ᴛʜᴇ ʀɪɢʜᴛ ᴀɴɢʟᴇ ᴛᴏ OA.
Answers
Answer:
Vᴇᴄᴛᴏʀ ᴇϙᴜᴀᴛɪᴏɴ ᴏғ ᴀ ᴘʟᴀɴᴇ ᴡʜᴇʀᴇ ɴ→ɴ→ ɪs ᴛʜᴇ ɴᴏʀᴍᴀʟ ᴛᴏ ᴛʜᴇ ᴘʟᴀɴᴇ ᴀɴᴅ ᴘᴀssɪɴɢ ᴛʜʀᴏᴜɢʜ ᴀ→ᴀ→ ɪs (ʀ→−ᴀ→).ɴ→=0(ʀ→−ᴀ→).ɴ→=0
Iғ (ᴀ,ʙ,ᴄ)(ᴀ,ʙ,ᴄ) ᴀʀᴇ ᴅɪʀᴇᴄᴛɪᴏɴ ʀᴀᴛɪᴏɴ ᴏғ ɢɪᴠᴇɴ ᴠᴇᴄᴛᴏʀ, ᴛʜᴇɴ ɪᴛs ᴅɪʀᴇᴄᴛɪᴏɴ ᴄᴏsɪɴᴇs ᴀʀᴇ (ᴀᴀ2+ʙ2+ᴄ2−−−−−−−−√,ʙᴀ2+ʙ2+ᴄ2−−−−−−−−√,ᴄᴀ2+ʙ2+ᴄ2−−−−−−−−√)(ᴀᴀ2+ʙ2+ᴄ2,ʙᴀ2+ʙ2+ᴄ2,ᴄᴀ2+ʙ2+ᴄ2)
Tʜᴇ ɢɪᴠᴇɴ ᴘᴏɪɴᴛs ᴀʀᴇ A(ᴀ,ʙ,ᴄ)A(ᴀ,ʙ,ᴄ) ᴀɴᴅ O(0,0,0)O(0,0,0)
OA−→−=(ᴀɪ^+ʙᴊ^+ᴄᴋ^)OA→=(ᴀɪ^+ʙᴊ^+ᴄᴋ^)
Wʜᴇʀᴇ ᴀ,ʙ,ᴄᴀ,ʙ,ᴄ ᴀʀᴇ ᴛʜᴇ ᴅɪʀᴇᴄᴛɪᴏɴ ʀᴀᴛɪᴏs
Tʜᴇ ᴅɪʀᴇᴄᴛɪᴏɴ ᴄᴏsɪɴᴇs ᴏғ OAOA ᴀʀᴇ
(ᴀᴀ2+ʙ2+ᴄ2−−−−−−−−√,ʙᴀ2+ʙ2+ᴄ2−−−−−−−−√,ᴄᴀ2+ʙ2+ᴄ2−−−−−−−−√)(ᴀᴀ2+ʙ2+ᴄ2,ʙᴀ2+ʙ2+ᴄ2,ᴄᴀ2+ʙ2+ᴄ2)
Nᴏᴡ ɪᴛ ɪs ɢɪᴠᴇɴ ᴛʜᴀᴛ ᴛʜᴇ ᴘʟᴀɴᴇ ɪᴅ ᴘᴇʀᴘᴇɴᴅɪᴄᴜʟᴀʀ ᴛᴏ OAOA
Hᴇɴᴄᴇ ɴ→=OA−→−=ᴀɪ^+ʙᴊ^+ᴄᴋ^ɴ→=OA→=ᴀɪ^+ʙᴊ^+ᴄᴋ^
Tʜᴇʀᴇғᴏʀᴇ ᴇϙᴜᴀᴛɪᴏɴ ᴏғ ᴛʜᴇ ᴘʟᴀɴᴇ ɪs(ʀ→−ᴀ→).ɴ→=0(ʀ→−ᴀ→).ɴ→=0
Sᴜʙsᴛɪᴛᴜᴛɪɴɢ ᴛʜᴇ ʀᴇsᴘᴇᴄᴛɪᴠᴇ ᴠᴇᴄᴛᴏʀs ᴡᴇ ɢᴇᴛ,
Eϙᴜᴀᴛɪᴏɴ ᴏғ ᴛʜᴇ ᴘʟᴀɴᴇ ɪs [ʀ→−(ᴀɪ^+ʙᴊ^+ᴄᴋ^)].[ᴀɪ^+ʙᴊ^+ᴄᴋ^]=0
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Answer:
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