Three Vectors, One Strange Identity

★ 8/10Cauchy-SchwarzAM-GMinequalityvectors

Problem

Plane vectors a⃗,b⃗,c⃗\vec a, \vec b, \vec c satisfy ∣b⃗∣⋅∣c⃗∣=1|\vec b|\cdot|\vec c| = 1 and

∣3a⃗−b⃗−c⃗∣=∣a⃗⋅b⃗∣⋅∣c⃗∣.\left|3\vec a - \vec b - \vec c\right| = \left|\vec a \cdot \vec b\right|\cdot\left|\vec c\right|.

Find the minimum of −3∣a⃗∣2+2∣b⃗∣2+∣c⃗∣2-3|\vec a|^2 + 2|\vec b|^2 + |\vec c|^2, and the value of ∣c⃗∣|\vec c| at which it is attained.

Answer

Solution

Difficulty8/10
TopicsCauchy-Schwarz, AM-GM, inequality, vectors

Whiteboard

Your sketch is saved only in this browser. To share it, export your drawing as an image (whiteboard menu → Export as → PNG), then upload that image in the comments below.

Discussion

Ask questions, share alternate solutions, and use LaTeX freely.

0 comments
Log in to join the discussion.

No comments yet.