Question

What is the value of (A vector + B vector) • (A vector x B vector) ?
(1) o
(2) A2 - B2
(3) A2 + B2 + 2AB
(4) none of these

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Answer 1

Answer:

ᴛʜɪs ɪs ᴀᴄᴛᴜᴀʟʟʏ ϙᴜɪᴛᴇ ᴀ ɢᴏᴏᴅ ϙᴜᴇsᴛɪᴏɴ, ʙᴇᴄᴀᴜsᴇ ɪᴛ ᴜsᴇs ᴀ ᴠᴇʀʏ ɪᴍᴘᴏʀᴛᴀɴᴛ ᴘʀᴏᴘᴇʀᴛʏ ᴏғ ᴛʜᴇ ᴄʀᴏss ᴘʀᴏᴅᴜᴄᴛ ᴀɴᴅ ᴡᴏʀᴋs ᴏᴜᴛ ɴɪᴄᴇʟʏ ᴀᴛ ᴛʜᴇ ᴇɴᴅ.

ᴛʜᴇ ɪᴍᴘᴏʀᴛᴀɴᴛ ᴘʀᴏᴘᴇʀᴛʏ ᴏғ ᴄʀᴏss ᴘʀᴏᴅᴜᴄᴛs ɪs:

ᴠ×ᴜ=−ᴜ×ᴠᴠ×ᴜ=−ᴜ×ᴠ

ᴛʜᴇ ᴡᴏʀᴅ ғᴏʀ ᴛʜɪs ɪs ᴀɴᴛɪsʏᴍᴍᴇᴛʀɪᴄ; ᴛʜᴇ “sʏᴍᴍᴇᴛʀɪᴄ” ᴘᴀʀᴛ ᴄᴏᴍᴇs ғʀᴏᴍ sᴡᴀᴘᴘɪɴɢ ᴛʜᴇ ᴛᴡᴏ ᴠᴇᴄᴛᴏʀs, ᴀɴᴅ ᴛʜᴇ “ᴀɴᴛɪ” ᴘᴀʀᴛ ᴄᴏᴍᴇs ғʀᴏᴍ ᴛʜᴇ ғᴀᴄᴛ ɪᴛ ɢɪᴠᴇs ʏᴏᴜ ᴛʜᴇ ᴍɪɴᴜs sɪɢɴ.

ᴡᴇ ᴡɪʟʟ ɴᴏᴛ ᴏɴʟʏ ɴᴇᴇᴅ ᴛʜɪs ᴛᴏ ʜᴇʟᴘ ᴀɴsᴡᴇʀ ᴛʜᴇ ϙᴜᴇsᴛɪᴏɴ, ᴡᴇ ᴄᴀɴ ᴀʟsᴏ ᴜsᴇ ɪᴛ ᴛᴏ ᴅᴇʀɪᴠᴇ sᴏᴍᴇᴛʜɪɴɢ ᴇʟsᴇ ᴡᴇ ᴡɪʟʟ ɴᴇᴇᴅ:

ᴜ×ᴜ=−ᴜ×ᴜᴜ×ᴜ=−ᴜ×ᴜ

ᴜ×ᴜ=0ᴜ×ᴜ=0

ɴᴏᴡ, ɪғ ᴡᴇ ᴇxᴘᴀɴᴅ ᴏᴜᴛ ᴛʜᴇ ʙʀᴀᴄᴋᴇᴛs ɪɴ ᴏᴜʀ ᴏʀɪɢɪɴᴀʟ ᴇxᴘʀᴇssɪᴏɴ, ᴡᴇ ᴡɪʟʟ ғɪɴᴅ:

(ᴀ−ʙ)×(ᴀ+ʙ)=ᴀ×ᴀ+ᴀ×ʙ−ʙ×ᴀ−ʙ×ʙ(ᴀ−ʙ)×(ᴀ+ʙ)=ᴀ×ᴀ+ᴀ×ʙ−ʙ×ᴀ−ʙ×ʙ

ᴛʜᴇ ᴛᴇʀᴍs ᴀ×ᴀᴀ×ᴀ ᴀɴᴅ ʙ×ʙʙ×ʙ ᴀʀᴇ ᴢᴇʀᴏ, sᴏ ᴡᴇ ᴄᴀɴ sɪᴍᴘʟʏ ʀᴇᴍᴏᴠᴇ ᴛʜᴇᴍ:

(ᴀ−ʙ)×(ᴀ+ʙ)=ᴀ×ʙ−ʙ×ᴀ(ᴀ−ʙ)×(ᴀ+ʙ)=ᴀ×ʙ−ʙ×ᴀ

ᴡᴇ ᴄᴀɴ ᴛʜᴇɴ sᴡᴀᴘ ᴛʜᴇ ʙʙ ᴀɴᴅ ᴀᴀ ᴠᴇᴄᴛᴏʀs ᴀʀᴏᴜɴᴅ ɪɴ ᴛʜᴇ sᴇᴄᴏɴᴅ ᴛᴇʀᴍ ᴜsɪɴɢ ᴛʜᴇ ᴀɴᴛɪsʏᴍᴍᴇᴛʀɪᴄ ᴘʀᴏᴘᴇʀᴛʏ:

(ᴀ−ʙ)×(ᴀ+ʙ)=ᴀ×ʙ+ᴀ×ʙ=2ᴀ ×ʙ

Explanation:

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