题目内容

Using the Rodrigues Rotation Formula, what effect does negating the axis of rotation have?

A. \( R(-\vec{a},\theta) = R(\vec{a},\theta) \)
B. \( R(-\vec{a},\theta) = R(\vec{a},\theta+\pi) \)
C. \( R(-\vec{a},\theta) = -R(\vec{a},\theta) \)
D. \( R(-\vec{a},\theta) = R(\vec{a},-\theta) \)
E. The result is unpredictable.
F. None of the above.

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Given a R2 vector， i⃗ =(2.05.0), 对这个向量施加一个350.0度的二维旋转，结果是什么？换句话说,j⃗ =R(350.0)(2.05.0)写出变换后向量j⃗ 的坐标j⃗ x=? j⃗ y=?

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Given two column vectors \( \vec{i} = \left( \begin{array}{c} i_x \\ i_y \\ i_z \end{array} \right) \) and \( \vec{j}= \left( \begin{array}{c} j_x \\ j_y \\ j_z \end{array} \right) \), what is the matrix multiplication expression equivalent to finding the dot product between the two vectors, \( \vec{i} \cdot \vec{j} \) ？

A. \( \vec{i}\ \vec{j} \)
B. \( \vec{i}\ \vec{j}^T \)
C. \( \vec{j}^T\ \vec{i} \)
D. \( \vec{j}\ \vec{i} \)

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Given that \( \vec{a} \), \( \vec{b} \), \( \vec{c} \) define an orthonormal basis, which of the following cannot be true?

A. \( \lVert\vec{a}\rVert = \vec{b} \cdot \vec{c} \)
B. \( \lVert\vec{a}\rVert = \lVert\vec{b}\rVert = \lVert\vec{c}\rVert = 1 \)
C. \( \vec{a}\cdot\vec{b} = \vec{a}\cdot\vec{c} = 0 \)
D. \( \vec{c} = \vec{a} \times \vec{b} \)
E. \( \vec{a}\cdot\vec{a} = \vec{b}\cdot\vec{b} = \vec{c}\cdot\vec{c} = 1 \)

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In a right-handed coordinate system, what is \( \vec{x}\times\vec{z} \) ?

A. \( \vec{y} \)
B. \( -\vec{y} \)
C. \( \vec{x} \)
D. \( -\vec{x} \)
E. \( \vec{z} \)
F. \( -\vec{z} \)

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Using the Rodrigues Rotation Formula, what effect does negating the axis of rotation have?

A. \( R(-\vec{a},\theta) = R(\vec{a},\theta) \) B. \( R(-\vec{a},\theta) = R(\vec{a},\theta+\pi) \) C. \( R(-\vec{a},\theta) = -R(\vec{a},\theta) \) D. \( R(-\vec{a},\theta) = R(\vec{a},-\theta) \) E. The result is unpredictable. F. None of the above.

Given a R2 vector， i⃗ =(2.05.0), 对这个向量施加一个350.0度的二维旋转，结果是什么？换句话说,j⃗ =R(350.0)(2.05.0)写出变换后向量j⃗ 的坐标j⃗ x=?______ j⃗ y=?______

Given that \( \vec{a} \), \( \vec{b} \), \( \vec{c} \) define an orthonormal basis, which of the following cannot be true?

In a right-handed coordinate system, what is \( \vec{x}\times\vec{z} \) ?

A. \( R(-\vec{a},\theta) = R(\vec{a},\theta) \)
B. \( R(-\vec{a},\theta) = R(\vec{a},\theta+\pi) \)
C. \( R(-\vec{a},\theta) = -R(\vec{a},\theta) \)
D. \( R(-\vec{a},\theta) = R(\vec{a},-\theta) \)
E. The result is unpredictable.
F. None of the above.

Given a R2 vector， i⃗ =(2.05.0), 对这个向量施加一个350.0度的二维旋转，结果是什么？换句话说,j⃗ =R(350.0)(2.05.0)写出变换后向量j⃗ 的坐标j⃗ x=? j⃗ y=?