Vectors & 3D
⏱ ~3-min readAceMark Guide
What this topic is really about
A cube has four body diagonals that connect opposite pairs of vertices through the interior of the cube. Option D is incorrect because 8 is the total number of vertices, not the number of pairs connecting them. Option C is incorrect because 6 represents the number of faces, which have face diagonals rather than space diagonals.
Two non-zero vectors a and b are perpendicular if their dot product is zero.. The dot product of two vectors is given by the product of their magnitudes and the cosine of the angle between them.
See the mechanism
The dot product of two vectors is given by the product of their magnitudes and the cosine of the angle between them. A diagram for this topic isn't available yet — the worked example below walks the same reasoning step by step.
An exam-style question, fully explained
If a · b = 0 for non-zero vectors a and b, they are:
- Identify what the question tests: If a · b = 0 for non-zero vectors a and b, they are:.
- The dot product of two vectors is given by the product of their magnitudes and the cosine of the angle between them.
- Since the vectors are non-zero, a dot product of zero implies the cosine of the angle is zero, meaning they are perpendicular.
- Option A is incorrect because parallel vectors would yield a non-zero dot product.
- Why it matters: The dot product of two vectors is given by the product of their magnitudes and the cosine of the angle between them. Since the vectors are non-zero, a dot product of zero implies the cosine of the angle is zero, meaning they are perpendicular. This is because the cosine of 90 degrees is zero.
Traps the examiner sets
- Many students confuse the concept of parallel and perpendicular vectors, and some may think that a zero dot product implies the vectors are parallel or equal, which is incorrect.
- Option A is incorrect because parallel vectors would yield a non-zero dot product.
- Option D is incorrect because it simply adds the components together instead of using the Euclidean distance formula.
- Option D is incorrect because 8 is the total number of vertices, not the number of pairs connecting them.
Test your recall
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