A unit vector has
Answer: A
It is obtained by dividing a vector by its own magnitude, which strips the size and keeps the orientation. Writing a vector as a magnitude times a unit vector separates cleanly what is geometrical from what is merely a number.
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The dot product of two vectors is
Answer: B
Correct answer: a scalar equal to the product of their magnitudes and the cosine of the angle between them
\[ \mathbf{A}\cdot\mathbf{B} = AB\cos\theta. \] The result is a number, not a vector, and it measures how much of one vector lies along the other - which is why it appears wherever a projection or a component is wanted, as in work done or flux through a surface.
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The cross product of two vectors is a vector whose direction is
Answer: C
The right hand rule fixes which of the two perpendicular senses is taken, which is why \( \mathbf{A} \times \mathbf{B} = -\mathbf{B} \times \mathbf{A} \). Its magnitude is the area of the parallelogram the two vectors span.
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If \( \mathbf{A} \cdot \mathbf{B} = 0 \) for two non-zero vectors, then they are
Answer: D
The dot product contains the cosine of the angle, which is zero only at ninety degrees for non-zero vectors. This is the standard test for orthogonality and it is how the unit vectors of any coordinate system are shown to be mutually perpendicular.
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The cross product of two vectors of finite length is found to vanish. This tells us that the pair must be
Answer: A
Correct answer: parallel or antiparallel
\[ |\mathbf{A}\times\mathbf{B}| = AB\sin\theta, \] which vanishes when \( \sin\theta = 0 \), that is at 0 or 180 degrees. Perpendicular vectors give the maximum cross product and a zero dot product, which is the opposite case and the trap here.
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The scalar triple product \( \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) \) represents
Answer: B
The cross product gives the base area and the dot product projects the third vector onto the normal to that base, giving base times height. The product is unchanged by a cyclic permutation of the three vectors, and it vanishes when they are coplanar.
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The vector triple product identity states that \( \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) \) equals
Answer: C
The mnemonic is that the result is the middle vector times the dot of the outer two, minus the last vector times the dot of the first two. The product is not associative, so the brackets cannot be moved without changing the answer.
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Two vectors are equal only if
Answer: D
Correct answer: their magnitudes and directions are both the same
A vector is specified by magnitude and direction and by nothing else, so equality requires both. In particular it does not require a common point of application: a free vector may be translated anywhere without becoming a different vector.
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The three variables of the cylindrical coordinate system are
Answer: A
Correct answer: \( \rho, \phi, z \)
Cylindrical coordinates use a radial distance from the axis, an azimuthal angle round it, and a height along it - so the system is a polar plane extended in the third direction. The set \( r, \theta, \phi \) belongs to spherical coordinates, where all three are measured from the origin.
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A point is to be located by a distance from the origin together with two angles. The system being used has the variables
Answer: B
Correct answer: \( r, \theta, \phi \)
One distance and two angles is the spherical arrangement: \( r \) from the origin, \( \theta \) measured down from the polar axis, and \( \phi \) round it. Cylindrical coordinates use two distances and one angle, and Cartesian three distances - so the count of angles identifies the system at once.
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