Define scalar triple product. How it can be expressed as a determinant.
Answers
Answer:
The scalar triple product of three vectors a, b, and c is (a×b)⋅c. ... The scalar triple product is important because its absolute value |(a×b)⋅c| is the volume of the parallelepiped spanned by a, b, and c (i.e., the parallelepiped whose adjacent sides are the vectors a, b, and c).
Explanation:
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Answer:
ʜᴇʀᴇ ɪs ʏᴏᴜʀ ᴀɴsᴡᴇʀ ⬇️
Explanation:
➡️ᴛʜᴇ sᴄᴀʟᴀʀ ᴛʀɪᴘʟᴇ ᴘʀᴏᴅᴜᴄᴛ ᴏғ ᴛʜʀᴇᴇ ᴠᴇᴄᴛᴏʀs ᴀ, ʙ, ᴀɴᴅ ᴄ ɪs (ᴀ×ʙ)⋅ᴄ.
➡️ᴛʜᴇ sᴄᴀʟᴀʀ ᴛʀɪᴘʟᴇ ᴘʀᴏᴅᴜᴄᴛ ɪs ɪᴍᴘᴏʀᴛᴀɴᴛ ʙᴇᴄᴀᴜsᴇ ɪᴛs ᴀʙsᴏʟᴜᴛᴇ ᴠᴀʟᴜᴇ |(ᴀ×ʙ)⋅ᴄ| ɪs ᴛʜᴇ ᴠᴏʟᴜᴍᴇ ᴏғ ᴛʜᴇ ᴘᴀʀᴀʟʟᴇʟᴇᴘɪᴘᴇᴅ sᴘᴀɴɴᴇᴅ ʙʏ ᴀ, ʙ, ᴀɴᴅ ᴄ (ɪ.ᴇ., ᴛʜᴇ ᴘᴀʀᴀʟʟᴇʟᴇᴘɪᴘᴇᴅ ᴡʜᴏsᴇ ᴀᴅᴊᴀᴄᴇɴᴛ sɪᴅᴇs ᴀʀᴇ ᴛʜᴇ ᴠᴇᴄᴛᴏʀs ᴀ, ʙ, ᴀɴᴅ ᴄ).
➡️ᴛᴏ ᴅᴇᴛᴇʀᴍɪɴᴇ ᴛʜᴇ ғᴏʀᴍᴜʟᴀ ғᴏʀ ᴛʜᴇ sᴄᴀʟᴀʀ ᴛʀɪᴘʟᴇ ᴘʀᴏᴅᴜᴄᴛ, ᴛʜᴇ ᴄʀᴏss ᴘʀᴏᴅᴜᴄᴛ ᴏғ ᴛᴡᴏ ᴠᴇᴄᴛᴏʀs ɪs ᴄᴀʟᴄᴜʟᴀᴛᴇᴅ ғɪʀsᴛ.
➡️ ᴀғᴛᴇʀ ᴛʜᴀᴛ, ᴛʜᴇ ᴅᴏᴛ ᴘʀᴏᴅᴜᴄᴛ ᴏғ ᴛʜᴇ ʀᴇᴍᴀɪɴɪɴɢ ᴠᴇᴄᴛᴏʀ ᴡɪᴛʜ ᴛʜᴇ ʀᴇsᴜʟᴛᴀɴᴛ ᴠᴇᴄᴛᴏʀ ɪs ᴄᴀʟᴄᴜʟᴀᴛᴇᴅ. ɪғ ᴛʜᴇ ᴛʀɪᴘʟᴇ ᴘʀᴏᴅᴜᴄᴛ ʀᴇsᴜʟᴛs ᴛᴏ ʙᴇ ᴢᴇʀᴏ, ᴛʜᴇɴ ɪᴛ sᴜɢɢᴇsᴛs ᴛʜᴀᴛ ᴏɴᴇ ᴏғ ᴛʜᴇ ᴛʜʀᴇᴇ ᴠᴇᴄᴛᴏʀs ᴛᴀᴋᴇɴ ɪs ᴏғ ᴢᴇʀᴏ ᴍᴀɢɴɪᴛᴜᴅᴇs.