9709 · 3.7
Vectors flashcards
Revision flashcards for Cambridge 9709 Vectors (syllabus 3.7). Flip, recall, then mark a real past-paper question.
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What is a position vector?
A vector from the origin O to a point P, denoted $\vec{OP}$ or simply $\mathbf{p}$. It defines the coordinates of point P.
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What is the formula for the magnitude of a 3D vector $\mathbf{v} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$?
The magnitude is its length, given by $|\mathbf{v}| = \sqrt{x^2 + y^2 + z^2}$.
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How do you find a unit vector in the direction of vector $\mathbf{a}$?
Divide the vector by its magnitude: $\hat{\mathbf{a}} = \frac{\mathbf{a}}{|\mathbf{a}|}$. A unit vector has a magnitude of 1.
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What is the scalar product of $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$ and $\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k}$?
In component form, it is $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$.
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What is the geometric definition of the scalar product?
$\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta$, where $\theta$ is the angle between the vectors when placed tail-to-tail.
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What is true about the scalar product of two non-zero perpendicular vectors?
Their scalar product is zero. $\mathbf{a} \cdot \mathbf{b} = 0$ because $\cos(90^\circ) = 0$.
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What is the vector equation of a line?
$\mathbf{r} = \mathbf{a} + \lambda\mathbf{d}$, where $\mathbf{a}$ is the position vector of a point on the line and $\mathbf{d}$ is the direction vector of the line.
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How do you find the angle between two intersecting lines, $l_1$ and $l_2$?
Find the angle $\theta$ between their direction vectors $\mathbf{d}_1$ and $\mathbf{d}_2$ using the scalar product: $\cos\theta = \frac{|\mathbf{d}_1 \cdot \mathbf{d}_2|}{|\mathbf{d}_1||\mathbf{d}_2|}$. The modulus in the numerator ensures you find the acute angle.
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Common Trap: What is the vector $\vec{AB}$ in terms of position vectors $\mathbf{a}$ and $\mathbf{b}$?
The vector from point A to point B is $\vec{AB} = \mathbf{b} - \mathbf{a}$. A common mistake is to calculate it as $\mathbf{a} - \mathbf{b}$ or $\mathbf{a} + \mathbf{b}$.
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When are two lines $\mathbf{r}_1 = \mathbf{a}_1 + \lambda\mathbf{d}_1$ and $\mathbf{r}_2 = \mathbf{a}_2 + \mu\mathbf{d}_2$ parallel?
When their direction vectors are scalar multiples of each other, i.e., $\mathbf{d}_1 = k\mathbf{d}_2$ for some scalar $k$.
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What does it mean if two lines are 'skew'?
They are not parallel and they do not intersect. This can only happen in 3D or higher dimensions.