9231 · 1.6
Vectors flashcards
Revision flashcards for Cambridge 9231 Vectors (syllabus 1.6). Flip, recall, then mark a real past-paper question.
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What is the vector product $\mathbf{a} \times \mathbf{b}$?
A vector that is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$. Its direction is given by the right-hand rule and its magnitude is $|\mathbf{a}||\mathbf{b}|\sin\theta$.
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How do you calculate $\mathbf{a} \times \mathbf{b}$ using components?
As the determinant of a matrix: $\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$.
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What is the geometric meaning of the magnitude $|\mathbf{a} \times \mathbf{b}|$?
It is the area of the parallelogram defined by the vectors $\mathbf{a}$ and $\mathbf{b}$.
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How are $\mathbf{a} \times \mathbf{b}$ and $\mathbf{b} \times \mathbf{a}$ related?
They are anti-commutative: $\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})$. The order of multiplication matters.
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What is the scalar triple product of vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$?
It is the scalar value given by $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$.
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How do you calculate the scalar triple product using components?
As the determinant of the matrix formed by the component of the vectors: $[\mathbf{a}, \mathbf{b}, \mathbf{c}] = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$.
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What is the geometric meaning of the absolute value of the scalar triple product, $|\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|$?
It is the volume of the parallelepiped defined by the vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$.
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What does it mean if the scalar triple product $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0$?
The three vectors are coplanar (they lie in the same plane), and the volume of the parallelepiped they form is zero.
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How do you find the area of a triangle with vertices A, B, and C?
Find two vectors representing sides, e.g., $\vec{AB}$ and $\vec{AC}$. The area is $\frac{1}{2}|\vec{AB} \times \vec{AC}|$. Don't forget the factor of $\frac{1}{2}$.
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How do you find the volume of a tetrahedron with one vertex at the origin and others at positions $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$?
The volume is $\frac{1}{6}|\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|$. It's one-sixth of the parallelepiped's volume.
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What is a common mistake when finding the shortest distance between skew lines?
Incorrectly identifying the vector that connects the two lines. It must be a vector from a specific point on line 1 to a specific point on line 2, e.g., $\mathbf{p}_2 - \mathbf{p}_1$.
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What is the vector product of a vector with itself, $\mathbf{a} \times \mathbf{a}$?
The zero vector, $\mathbf{0}$. This is because the angle $\theta$ between the vectors is 0, and $\sin(0) = 0$.