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If a, b, c are non-coplanar vectors and...

If a, b, c are non-coplanar vectors and r is a unit vector, then `|(r.a) (b xx c) +(r.b) (c xx a) + (r.c) ( a xxb)| =`

A

`[a bc]^(2)`

B

`|[abc]|`

C

1

D

none

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The correct Answer is:
To solve the problem, we need to find the expression: \[ |(r \cdot a)(b \times c) + (r \cdot b)(c \times a) + (r \cdot c)(a \times b)| \] where \( a, b, c \) are non-coplanar vectors and \( r \) is a unit vector. ### Step 1: Understanding the Components We know that \( r \) is a unit vector, which means \( |r| = 1 \). The terms \( r \cdot a \), \( r \cdot b \), and \( r \cdot c \) are scalars representing the projections of \( a, b, c \) onto the direction of \( r \). ### Step 2: Scalar Triple Product The expression can be interpreted as a combination of scalar triple products. The scalar triple product \( a \cdot (b \times c) \) gives the volume of the parallelepiped formed by the vectors \( a, b, c \). Since \( a, b, c \) are non-coplanar, this value is non-zero. ### Step 3: Rewrite the Expression We can rewrite the expression in terms of the scalar triple product: \[ |(r \cdot a)(b \times c) + (r \cdot b)(c \times a) + (r \cdot c)(a \times b)| \] This can be viewed as a linear combination of the vectors \( b \times c, c \times a, a \times b \) scaled by the respective dot products with \( r \). ### Step 4: Factor Out the Magnitude Using properties of determinants and the scalar triple product, we can factor out the magnitude of the scalar triple product: \[ |(r \cdot a)(b \times c) + (r \cdot b)(c \times a) + (r \cdot c)(a \times b)| = |r| \cdot |(a, b, c)| \] Since \( |r| = 1 \), we have: \[ |(r \cdot a)(b \times c) + (r \cdot b)(c \times a) + (r \cdot c)(a \times b)| = |(a, b, c)| \] ### Step 5: Conclusion The magnitude of the scalar triple product \( |(a, b, c)| \) is a non-zero value since \( a, b, c \) are non-coplanar. Therefore, the final answer is: \[ |(r \cdot a)(b \times c) + (r \cdot b)(c \times a) + (r \cdot c)(a \times b)| = 1 \] ### Final Answer \[ \boxed{1} \]
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