Scalars, Vectors, Dot & Cross Products

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Questions and Answers

Explain the difference between a scalar quantity and a vector quantity, providing one example of each that is not mentioned in the text.

A scalar quantity is described only by its magnitude, while a vector quantity requires both magnitude and direction. An example of a scalar quantity is electric charge and a vector quantity is torque.

Given two vectors, A⃗ and B⃗, describe how the angle θ between them affects the outcome of their dot product (A⃗⋅B⃗).

The dot product A⃗⋅B⃗ is equal to |A⃗||B⃗|cosθ. Therefore, as θ increases from 0° to 90°, cosθ decreases from 1 to 0, reducing the dot product's magnitude. When θ is 90°, the dot product is zero, indicating that the vectors are orthogonal.

Explain the significance of a unit vector and how it is derived from any given vector A⃗.

A unit vector indicates direction with a magnitude of one. It is derived by dividing the vector A⃗ by its magnitude |A⃗|, resulting in Â = A⃗/|A⃗|.

If you know both the dot product and cross product of two vectors, what information can you directly infer about the relationship between these vectors?

Knowing the dot product and cross product allows you to determine the cosine and sine of the angle between the vectors, respectively. This provides a complete understanding of their relative orientation in space. Signup and view all the answers

Describe the difference in the results when the del operator (∇) operates on a scalar field versus a vector field.

When the del operator acts on a scalar field, it results in the gradient, which is a vector field. When it acts on a vector field, it can result in either the divergence, which is a scalar field, or the curl, which is a vector field. Signup and view all the answers

Explain why emotions are not considered physical quantities according to the definition presented.

Emotions like happiness and sadness are not considered physical quantities because they cannot be measured numerically. Physical quantities must be quantifiable to define laws of physics. Signup and view all the answers

Given vectors A⃗ = 3î - 2ĵ + k̂ and B⃗ = -î + ĵ - 2k̂, determine if A⃗ and B⃗ are orthogonal. Justify your answer.

Two vectors are orthogonal if their dot product is zero. A⃗⋅B⃗ = (3*-1) + (-21) + (1-2) = -3 - 2 - 2 = -7. Since A⃗⋅B⃗ is not 0, A⃗ and B⃗ are not orthogonal. Signup and view all the answers

Explain why the order of vectors matters in the cross product (A⃗ × B⃗), but not in the dot product (A⃗⋅B⃗).

The cross product (A⃗ × B⃗) results in a vector perpendicular to both A⃗ and B⃗, with its direction determined by the right-hand rule. Reversing the order changes the direction of the resultant vector (A⃗ × B⃗ = -B⃗ × A⃗). The dot product (A⃗⋅B⃗) yields a scalar and only depends on the magnitudes of the vectors and the cosine of the angle between them, making it commutative. Signup and view all the answers

Describe a real-world scenario where understanding vector addition is crucial for accurate calculations or predictions.

In navigation, vector addition is essential for calculating the resultant displacement of an aircraft or ship. The velocity vectors of the aircraft and the wind (or the ship and the ocean current) must be added to determine the actual path and speed over the ground or water. Signup and view all the answers

Explain the difference between divergence and curl for a vector field. What does each tell you about the field's behavior?

Divergence measures the extent to which a vector field 'sources' or 'sinks' at a given point, resulting in a scalar field. Curl measures the 'rotation' or 'circulation' of a vector field at a point, resulting in another vector field. Signup and view all the answers

Flashcards

Scalar Quantity

A physical quantity fully described by its magnitude (size or quantity).

Physical Quantity

Any measurable aspect of the physical world.