Vectors, Products, and Subspaces Practice Quiz (10 questions)

1. Which description correctly defines the real plane $\mathbb{R}^2$?

A. The set of all ordered pairs of real numbers

B. The set of all vectors with two equal coordinates

C. The set of all ordered triples of real numbers

D. The set of all positive pairs of real numbers

Answer: (A)

Explanation

The real plane consists of ordered pairs $\left(x,y\right)$ in which both coordinates are real numbers. Ordered triples belong to $\mathbb{R}^3$, not $\mathbb{R}^2$.

2. Which three features characterize the geometric vector from point A to point B?

A. Length, slope, and distance from the origin

B. Magnitude, endpoint location, and coordinate system

C. Magnitude, direction, and orientation from A toward B

D. Initial coordinate, terminal coordinate, and midpoint

Answer: (C)

Explanation

A geometric vector from A to B is determined by its magnitude, direction, and orientation from the initial point toward the terminal point. The coordinates or midpoint can describe a placement but do not constitute the defining three features.

3. Which condition shows that w is a linear combination of u and v?

A. The vectors have equal magnitudes and point in the same direction

B. There are real scalars λ and μ such that $w=\lambda u+\mu v$

C. The vectors satisfy $w=u+v$ with both coefficients fixed at one

D. The vector w is formed by connecting the endpoints of u and v

Answer: (B)

Explanation

A linear combination permits arbitrary real scalar coefficients, so it has the form $w=\lambda u+\mu v$. Plain vector addition is a special case with coefficients equal to one, but it does not express the full definition.

4. If points A, B, and C form a chain of directed segments, which equation gives Chasles’ relation?

A. $\overrightarrow{AB}+\overrightarrow{AC}=\overrightarrow{BC}$

B. $\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$

C. $\overrightarrow{BA}+\overrightarrow{BC}=\overrightarrow{AC}$

D. $\overrightarrow{AC}+\overrightarrow{BC}=\overrightarrow{AB}$

Answer: (B)

Explanation

Following the displacement from A to B and then from B to C produces the direct displacement from A to C. The other equations reverse or combine directed segments in ways that do not represent this continuous path.

5. What is the primary purpose of the dot product between two vectors in Euclidean space?

A. To compute the area of a parallelogram.

B. To find the length of a vector.

C. To measure the angle between the vectors and determine orthogonality.

D. To identify if two vectors are linearly independent.

Answer: (C)

Explanation

The dot product helps measure the cosine of the angle between two vectors and indicates orthogonality when it equals zero. It does not directly measure length, area, or linear independence.

6. What is the defining characteristic of a basis in \\mathbb{R}^2?

A. Two non-collinear vectors that allow every vector in \\mathbb{R}^2 to be expressed as a unique linear combination.

B. A set of vectors that are all coplanar.

C. Any two vectors that are not scalar multiples of each other.

D. Two vectors that are orthogonal and of unit length.

Answer: (A)

Explanation

A basis in \\mathbb{R}^2 consists of two non-collinear vectors, which ensures that every vector in the plane can be uniquely written as a linear combination of these basis vectors. The other options do not guarantee unique representation or are not sufficient to form a basis.

7. How does the determinant of two vectors in the plane differ from their dot product in terms of the geometric information they provide?

A. The determinant is always positive and gives the area, while the dot product can be negative and measures the length of the projection.

B. The determinant and the dot product are essentially the same, both measuring the area of the parallelogram spanned by the vectors.

C. The determinant indicates whether the vectors are collinear, whereas the dot product indicates the angle's cosine and the magnitude of the projection.

D. The determinant measures the signed area of the parallelogram spanned by the vectors, while the dot product measures the cosine of the angle between them.

Answer: (C)

Explanation

The determinant of two vectors in the plane is zero if and only if they are collinear, and its absolute value gives the area of the parallelogram they span. The dot product, on the other hand, relates to the cosine of the angle between the vectors and their magnitudes, not the area.

8. What is the primary consequence of the cross product of two non-collinear vectors in space?

A. It computes the volume of the parallelepiped formed by the vectors, but does not produce a vector.

B. It produces a vector orthogonal to both original vectors, with magnitude equal to the area of the parallelogram they span.

C. It gives a vector parallel to the first vector, scaled by the sine of the angle between them.

D. It results in a scalar equal to the dot product of the vectors, indicating their angle.

Answer: (B)

Explanation

The cross product of two non-collinear vectors in space yields a vector orthogonal to both, with magnitude equal to the area of the parallelogram they span. Unlike the scalar triple product, it does not directly give volume, but the norm relates to the parallelogram's area.

9. How can the mixed product of three vectors in space be used to determine the volume of a parallelepiped?

A. The absolute value of the scalar triple product gives the volume.

B. The scalar triple product is unrelated to volume but measures the parallelepiped's height.

C. The scalar triple product directly measures the surface area of the parallelepiped.

D. The mixed product equals zero if the vectors are coplanar, indicating zero volume.

Answer: (A)

Explanation

The absolute value of the scalar triple product of three vectors equals the volume of the parallelepiped they span. If the vectors are coplanar, the mixed product is zero, indicating no volume.

10. What is a key feature of a normal vector to a subspace in Euclidean space?

A. A normal vector is parallel to the subspace.

B. A normal vector has zero magnitude.

C. A normal vector is orthogonal to every vector in the subspace.

D. A normal vector lies within the subspace.

Answer: (C)

Explanation

A normal vector is characterized by its orthogonality to all vectors in the subspace it is normal to. It is not necessarily within the subspace itself, nor does it have zero magnitude or parallelism to the subspace.