Standard +0.3 This is a straightforward 3D vector problem requiring students to establish coordinates, form two vectors, apply the scalar product formula, and find an angle. While it involves multiple steps and 3D visualization, it uses standard techniques (dot product for angles) with no conceptual surprises, making it slightly easier than average for A-level.
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The diagram shows a solid figure \(O A B C D E F G\) with a horizontal rectangular base \(O A B C\) in which \(O A = 8\) units and \(A B = 6\) units. The rectangle \(D E F G\) lies in a horizontal plane and is such that \(D\) is 7 units vertically above \(O\) and \(D E\) is parallel to \(O A\). The sides \(D E\) and \(D G\) have lengths 4 units and 2 units respectively. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively. Use a scalar product to find angle \(O B F\), giving your answer in the form \(\cos ^ { - 1 } \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers.
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The diagram shows a solid figure $O A B C D E F G$ with a horizontal rectangular base $O A B C$ in which $O A = 8$ units and $A B = 6$ units. The rectangle $D E F G$ lies in a horizontal plane and is such that $D$ is 7 units vertically above $O$ and $D E$ is parallel to $O A$. The sides $D E$ and $D G$ have lengths 4 units and 2 units respectively. Unit vectors $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$ are parallel to $O A , O C$ and $O D$ respectively. Use a scalar product to find angle $O B F$, giving your answer in the form $\cos ^ { - 1 } \left( \frac { a } { b } \right)$, where $a$ and $b$ are integers.\\
\hfill \mbox{\textit{CAIE P1 2018 Q6 [6]}}