Let a, b satisfy | a+ b|^2=| a|^2+2| b|^2, a b=3, | a b|^2=75. · questionr

questionrEarly access

Question 219 of 379MediumNUMERICAL

Let

\vec a,\vec b

satisfy

|\vec a+\vec b|^2=|\vec a|^2+2|\vec b|^2

,

\vec a\cdot\vec b=3

,

|\vec a\times\vec b|^2=75

. Then

|\vec a|^2

is \_\_\_\_\_\_\_. Answer: 14

Solutions · 1

Official solution

|\vec a+\vec b|^2 = |\vec a|^2+|\vec b|^2+2\vec a\cdot\vec b = |\vec a|^2+2|\vec b|^2 \implies |\vec b|^2=2\vec a\cdot\vec b=6

\[|\vec a\times\vec b|^2 = |\vec a|^2|\vec b|^2-(\vec a\cdot\vec b)^2 \implies 75 = 6|\vec a|^2-9 \implies |\vec a|^2=14\]

Answer the question to unlock