If a vector $\alpha$ lies in the plane of $\beta$ and $\gamma$, then (1) $[\alpha,\beta,\gamma]=1$ (2) $[\alpha,\beta,\gamma]=-1$ (3) $[\alpha,\beta,\gamma]=0$ (4) $[\alpha,\beta,\gamma]=2$

If a vector α lies in the plane of β and γ, then (1) [α,β,γ]=1 (2) [α,β,γ]=-1 (3) [α,β,γ]=0 (4) [α,β,γ]=2

Exercise 6.10 · Q2

\alpha

(1)

[\alpha,\beta,\gamma]=1

(2)

[\alpha,\beta,\gamma]=-1

(3)

[\alpha,\beta,\gamma]=0

(4)

[\alpha,\beta,\gamma]=2

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✓ Free question

" $\alpha$ lies in the plane of $\beta,\gamma$ " is exactly the geometric definition of $\alpha,\beta,\gamma$ being coplanar.

Step 1. Translate the hypothesis. $\alpha$ lying in the plane spanned by $\beta,\gamma$ means $\alpha,\beta,\gamma$ are three coplanar vectors.

Step 2. Apply the coplanarity test (Theorem 6.4). Three vectors are coplanar iff their scalar triple product is $0$ .

Step 3. Conclude. $[\alpha,\beta,\gamma]=0$ , option (3).

✓Final answer

[\alpha,\beta,\gamma]=\boxed0