International Baccalaureate Mathematics

Syllabus Content

The definition of the vector product of two vectors.
Properties of the vector product.
Geometric interpretation of |v×w|

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Vectors Vectors usually first make an appearance when pupils learn about transformations. A translation is best described with a vector written as a two by one matrix. Following that pupils learn how vectors can be used to prove geometric relationships in simple line diagrams. Ultimately vectors are studied as a major topic for A Level and International Baccalaureate courses where vectors in three dimensions are included.You can think of a vector as what is needed to 'carry' the point A to the point B. The Latin word vector means ‘carrier’ and was first used by 18th century astronomers investigating planet rotation around the Sun.

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Official Guidance, clarification and syllabus links:

The vector product is also known as the “cross product”.

$ v×w=|v||w| \sin \theta \; n$, where $ \theta $ is the angle between $v$ and $w$, and $n$ is the unit normal vector whose direction is given by the right-hand screw rule.

$v×w=-w×v$;

$u×(v+w)=u×v+u×w$;

$(kv)×w=k(v×w)$;

$v×v=0$.

For non-zero vectors $v×w=0$ is equivalent to the vectors being parallel.

Use of $ |v×w| $ to find the area of a parallelogram (and hence a triangle).

Formula Booklet AHL 3.16:

Vector product

$ \mathbf{v \times w} = \begin{pmatrix} v_2w_3-v_3w_2 \\ v_3w_1-v_1w_3 \\ v_1w_2-v_2w_1 \end{pmatrix}. \quad \text{where} \quad \mathbf{v} =\begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}, \quad \mathbf{w} = \begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix} $

$ \mathbf{|v \times w|} = \mathbf{|v||w|} \sin{ \theta} $, where $ \theta $ is the angle between $ \mathbf{v} $ and $ \mathbf{w} $

Area of a parallelogram

$A=|\mathbf{v \times w|}$ where $ \mathbf{v} $ and $ \mathbf{w} $ form two adjacent sides of a parallelogram

The vector product of two vectors is a binary operation that produces a vector perpendicular to both of the input vectors. It is also known as the cross product.

The formula for the vector product of two vectors $ \vec{a} $ and $\vec{b} $ is given by:

$$\vec{a} \times \vec{b} = \begin{pmatrix} a_{1} \\ a_{2} \\ a_{3} \end{pmatrix} \times \begin{pmatrix} b_{1} \\ b_{2} \\ b_{3} \end{pmatrix} = \begin{pmatrix} a_{2}b_{3} - a_{3}b_{2} \\ a_{3}b_{1} - a_{1}b_{3} \\ a_{1}b_{2} - a_{2}b_{1} \end{pmatrix}$$

For example, let $ \vec{a} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix} $ and $ \vec{b} = \begin{pmatrix} 4 \\ -1 \\ 5 \end{pmatrix} $. Then:

$$\vec{a} \times \vec{b} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix} \times \begin{pmatrix} 4 \\ -1 \\ 5 \end{pmatrix} = \begin{pmatrix} (3)(5) - (1)(-1) \\ (1)(4) - (2)(5) \\ (2)(-1) - (3)(4) \end{pmatrix} = \begin{pmatrix} 16 \\ -6 \\ -14 \end{pmatrix}$$

Therefore, $ \vec{a} \times \vec{b} = \begin{pmatrix} 16 \\ -6 \\ -14 \end{pmatrix} $.

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International Baccalaureate Mathematics

Geometry and Trigonometry

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