Quiz: Vectors and Vector Subspaces — 31 questions

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

A. The set of vectors with three components

B. The set of ordered pairs of real numbers

C. The set of ordered triples of real numbers

D. The set of positive real number pairs

Answer: (B)

Explanation

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

2. A point in real space $\mathbb{R}^3$ must be represented by which type of coordinate tuple?

A. An ordered triple of real numbers

B. A pair of vectors in the plane

C. A single nonnegative real number

D. An ordered pair of real numbers

Answer: (A)

Explanation

Real space is defined as the set of ordered triples $\left(x,y,z\right)$ with real coordinates. An ordered pair belongs to the real plane $\mathbb{R}^2$.

3. What three geometric features characterize the vector $\overrightarrow{AB}$ from two distinct points A and B?

A. Slope, intercept, and distance from the origin

B. Length, midpoint, and angle at the origin

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

D. Initial point, terminal point, and midpoint

Answer: (C)

Explanation

The vector $\overrightarrow{AB}$ has the magnitude of segment AB, the direction of line AB, and orientation from A toward B. Its midpoint and slope are not part of this defining trio.

4. Two geometric vectors can be equal even when they have different initial points provided that they share which properties?

A. The same magnitude, direction, and orientation

B. The same coordinates relative to one fixed origin

C. The same initial point, endpoint, and midpoint

D. The same supporting line and intersection point

Answer: (A)

Explanation

Equal vectors have identical magnitude, direction, and orientation, regardless of where they are positioned. Sharing an initial point is a property of coincident geometric segments, not a requirement for equal vectors.

5. How is the sum of two nonzero vectors with different directions represented geometrically?

A. By the longer side of the triangle formed by the vectors

B. By the midpoint of the segment joining their endpoints

C. By a vector obtained after reversing the second vector

D. By the diagonal of the parallelogram formed by the vectors

Answer: (D)

Explanation

Placing the two vectors as adjacent sides of a parallelogram makes their sum the diagonal from the common initial point. Reversing a vector changes its orientation rather than representing vector addition.

6. If a vector $\vec u$ is multiplied by a negative scalar $\lambda$, what happens to its orientation?

A. Its direction becomes perpendicular to the original

B. Its magnitude becomes zero

C. Its orientation is preserved

D. Its orientation is reversed

Answer: (D)

Explanation

A negative scalar reverses the orientation of the vector, while the magnitude is scaled by the scalar's absolute value. A zero scalar produces the zero vector, but a negative scalar does not.

7. Which expression shows that $\vec w$ is a linear combination of $\vec u$ and $\vec v$?

A. The vectors satisfy $\vec w=\vec u+\vec v$ with equal magnitudes

B. There is a real number $\lambda$ such that $\vec w=\lambda\vec u$

C. The vectors satisfy $\vec w\cdot\vec u=\vec v$ for some real coordinates

D. There are real numbers $\lambda$ and $\mu$ such that $\vec w=\lambda\vec u+\mu\vec v$

Answer: (D)

Explanation

A linear combination permits two real coefficients and has the form $\vec w=\lambda\vec u+\mu\vec v$. Restricting the expression to one coefficient does not capture the stated definition in general.

8. Which standard basis corresponds to an orthonormal frame in $\mathbb{R}^3$?

A. $\left(\vec i,\vec k\right)$

B. $\left(\vec j,\vec k\right)$

C. $\left(\vec i,\vec j\right)$

D. $\left(\vec i,\vec j,\vec k\right)$

Answer: (D)

Explanation

The standard basis of $\mathbb{R}^3$ contains the three vectors $\vec i$, $\vec j$, and $\vec k$. The pair $\left(\vec i,\vec j\right)$ is the standard basis for $\mathbb{R}^2$.

9. What is the magnitude of a vector with coordinates $\left(6,8\right)$ in $\mathbb{R}^2$?

A. $14$

B. $48$

C. $10$

D. $\sqrt{28}$

Answer: (C)

Explanation

Using the planar norm gives $\|\vec u\|=\sqrt{x^2+y^2}=\sqrt{6^2+8^2}=\sqrt{100}=10$. Adding the coordinates or multiplying them does not apply the magnitude formula.

10. If $A=(2,-1)$ and $B=(7,4)$ in $\mathbb{R}^2$, what is the coordinate vector $\overrightarrow{AB}$?

A. $(9,3)$

B. $(2,-1,7,4)$

C. $(-5,-5)$

D. $(5,5)$

Answer: (D)

Explanation

The coordinate vector is found by subtracting A's coordinates from B's coordinates: $\overrightarrow{AB}=(x_B-x_A,y_B-y_A)=(7-2,4-(-1))=(5,5)$. Reversing the subtraction would produce the vector from B to A.

11. Which condition distinguishes collinearity of two vectors from coplanarity of three vectors in $\mathbb{R}^3$?

A. Collinearity requires a zero dot product, whereas coplanarity requires equal lengths

B. Collinearity concerns determinants, whereas coplanarity concerns vector magnitudes

C. Collinearity uses a scalar multiple, whereas coplanarity uses a linear combination of two vectors

D. Collinearity uses a linear combination, whereas coplanarity uses a scalar multiple

Answer: (C)

Explanation

Two vectors are collinear when one is a scalar multiple of the other, while three vectors are coplanar when one can be expressed as a linear combination of the other two. A linear combination is therefore the relevant criterion for coplanarity, not collinearity.

12. Why do two non-collinear vectors form a basis of $\mathbb{R}^2$?

A. They provide a unique linear combination for every vector in the plane

B. They generate the plane even when one vector is redundant

C. They have equal lengths and point in perpendicular directions

D. They are scalar multiples of one another and share one direction

Answer: (A)

Explanation

Two non-collinear vectors span the plane and give every vector a unique linear combination, which is the defining basis property. A generating family can contain redundant vectors, but a basis cannot have that kind of redundancy.

13. If $\vec u=(2,-1,3)$ and $\vec v=(4,0,-2)$, what is their dot product?

A. 10

B. 2

C. -2

D. -8

Answer: (C)

Explanation

The three-dimensional dot product is computed by multiplying corresponding coordinates and adding: $2\times4+(-1)\times0+3\times(-2)=-2$. The result is a scalar, unlike a cross product, which produces a vector.

14. Two vectors have a dot product of zero; what can be concluded about them?

A. They point in the same direction

B. They are collinear

C. They are orthogonal

D. They have equal lengths

Answer: (C)

Explanation

Vectors are orthogonal if and only if their dot product is zero. Collinear vectors generally have a nonzero dot product unless a zero vector is involved, so collinearity is a different condition.

15. For nonzero vectors $\vec u=(1,2)$ and $\vec v=(2,0)$, what is the orthogonal projection of $\vec u$ onto $\vec v$?

A. $(0,1)$

B. $(1,0)$

C. $(1,2)$

D. $(2,0)$

Answer: (B)

Explanation

Using $p_{\vec v}(\vec u)=\frac{\vec u\cdot\vec v}{\|\vec v\|^2}\vec v$ gives $\frac{2}{4}(2,0)=(1,0)$. The projection must be a scalar multiple of $\vec v$, so it lies along the direction of $\vec v$.

16. Two nonzero vectors have lengths $3$ and $4$ and form an angle of $60^\circ$. What is their dot product?

A. 1

B. 6

C. 7

D. 12

Answer: (B)

Explanation

The angle formula gives $\vec u\cdot\vec v=\|\vec u\|\|\vec v\|\cos(60^\circ)=3\times4\times\frac12=6$. Multiplying the lengths without the cosine factor would incorrectly ignore the angle between the vectors.

17. For $\vec u=(3,1)$ and $\vec v=(2,4)$, what is $\det(\vec u,\vec v)$?

A. 14

B. 2

C. -10

D. 10

Answer: (D)

Explanation

The planar determinant is $a_1b_2-a_2b_1=3\times4-1\times2=10$. Its sign records orientation, so it is not the same calculation as adding coordinate products.

18. What is the area of the parallelogram spanned by $\vec u=(1,3)$ and $\vec v=(2,1)$?

A. 7

B. 2

C. 5

D. -5

Answer: (C)

Explanation

The determinant is $1\times1-3\times2=-5$, and the geometric area is its absolute value, $|-5|=5$. A signed determinant can be negative because it records orientation, but area is nonnegative.

19. Which description correctly characterizes the cross product of two non-collinear vectors in $\mathbb{R}^3$?

A. It is a vector parallel to both inputs, with length equal to their angle

B. It is a scalar equal to the sum of the corresponding coordinate products

C. It is a scalar measuring the signed area between the two input vectors

D. It is a vector perpendicular to both inputs, with direction set by the right-hand rule

Answer: (D)

Explanation

The cross product is perpendicular to both non-collinear input vectors, and its orientation follows the right-hand rule. The dot product, not the cross product, produces a scalar from two vectors.

20. What is the cross product of $\vec u=(1,2,3)$ and $\vec v=(4,5,6)$?

A. $(6,-3,-3)$

B. $(-3,6,-3)$

C. $(-3,-6,3)$

D. $(3,-6,3)$

Answer: (B)

Explanation

Applying the coordinate formula gives $\vec u\wedge\vec v=(2\times6-3\times5,3\times4-1\times6,1\times5-2\times4)=(-3,6,-3)$. The other choices contain sign or component-order errors.

21. Two vectors in $\mathbb{R}^3$ have lengths 4 and 5 and form an angle of $30^\circ$; what is the area of the parallelogram they span?

A. $10$

B. $20$

C. $40$

D. $\frac{5\sqrt{3}}{2}$

Answer: (A)

Explanation

The parallelogram area is the cross-product norm, so it equals $4\times5\times\sin(30^\circ)=10$. Multiplying the lengths without the sine factor would give the wrong value for a non-right angle.

22. What kind of quantity is the mixed product $\det(\vec u,\vec v,\vec w)=\vec u\cdot(\vec v\wedge\vec w)$?

A. A vector perpendicular to all three inputs

B. A scalar

C. A matrix with three rows

D. A pair of perpendicular vectors

Answer: (B)

Explanation

The cross product $\vec v\wedge\vec w$ is a vector, and taking its dot product with $\vec u$ produces a scalar. Confusing the mixed product with the cross product incorrectly assigns it a vector value.

23. Three vectors have mixed-product determinant $-12$; what volume does their parallelepiped have?

A. $-12$

B. $12$

C. $\frac{1}{12}$

D. $144$

Answer: (B)

Explanation

The parallelepiped volume is the absolute value of the mixed product, so $V=|-12|=12$. The negative determinant records orientation, not a negative physical volume.

24. Three vectors in $\mathbb{R}^3$ have determinant zero; what conclusion follows about them?

A. They do not form a basis of $\mathbb{R}^3$

B. They span a parallelepiped of unit volume

C. They form an orthonormal basis of $\mathbb{R}^3$

D. They are necessarily three pairwise perpendicular vectors

Answer: (A)

Explanation

Three vectors form a basis of $\mathbb{R}^3$ if and only if their mixed-product determinant is nonzero. A zero determinant means the vectors are linearly dependent and therefore cannot form a basis.

25. Which set describes $\operatorname{Span}(\vec u,\vec v)$?

A. All vectors of the form $\lambda\vec u+\mu\vec v$ with $\lambda,\mu\in\mathbb{R}$

B. All vectors perpendicular to both $\vec u$ and $\vec v$

C. The two listed vectors $\vec u$ and $\vec v$ without additional vectors

D. All vectors whose coordinates equal the coordinates of either input vector

Answer: (A)

Explanation

The span consists of every linear combination $\lambda\vec u+\mu\vec v$ with real coefficients. The listed vectors are generators, whereas their span can contain infinitely many additional vectors.

26. If $\vec w=2\vec u-3\vec v$, what happens to $\operatorname{Span}(\vec u,\vec v,\vec w)$ when $\vec w$ is removed?

A. The spanned subspace becomes the set of vectors perpendicular to $\vec u$

B. The spanned subspace becomes the line through $\vec w$

C. The spanned subspace remains unchanged

D. The spanned subspace becomes larger because one constraint is removed

Answer: (C)

Explanation

Because $\vec w$ is already a linear combination of $\vec u$ and $\vec v$, it adds no new direction to the span. Removing a redundant generator therefore leaves the spanned subspace unchanged.

27. What condition must a vector $\vec n$ satisfy to be normal to a subspace $S$?

A. It must belong to the generating family of $S$

B. It must have the same length as every vector in $S$

C. It must be parallel to at least one vector in $S$

D. It must be orthogonal to every vector in $S$

Answer: (D)

Explanation

A normal vector is orthogonal to every vector in the subspace. Being listed as a generator or parallel to one member of the subspace does not establish orthogonality to the entire subspace.

28. What geometric object is formed by the span of a nonzero vector $\vec u=(a,b)$ in $\mathbb{R}^2$?

A. An affine line that may be displaced from the origin

B. A vector line through the origin with direction vector $\vec u$

C. A vector plane generated by two independent directions

D. A single point determined by the endpoint of $\vec u$

Answer: (B)

Explanation

The span of a nonzero vector in $\mathbb{R}^2$ consists of all scalar multiples of that vector, forming a vector line through the origin. An affine line can fail to pass through the origin, so it is a different geometric object.

29. Which Cartesian equation represents the vector line generated by $\vec u=(3,2)$?

A. $3x+2y=0$

B. $2y+3x=1$

C. $3y-2x=0$

D. $2x-3y=1$

Answer: (C)

Explanation

For $\vec u=(a,b)$, the generated vector line has equation $ay-bx=0$; substituting $a=3$ and $b=2$ gives $3y-2x=0$. Equations with a nonzero constant describe affine lines rather than vector lines through the origin.

30. What condition on two vectors in $\mathbb{R}^3$ ensures that their span is a vector plane?

A. They are non-collinear vectors

B. They have equal lengths and opposite directions

C. They are scalar multiples of one another

D. They both have a zero coordinate

Answer: (A)

Explanation

Two non-collinear vectors provide two distinct directions, so their span is a vector plane in $\mathbb{R}^3$. Scalar multiples are collinear and span a line instead of a plane.

31. A plane has Cartesian equation $4x-3y+2z=0$. Which vector is normal to this plane?

A. $(-3,2,4)$

B. $(2,-3,4)$

C. $(4,3,2)$

D. $(4,-3,2)$

Answer: (D)

Explanation

For a plane written as $ax+by+cz=0$, the coefficient vector $(a,b,c)$ is a normal vector. Here the coefficients give $(4,-3,2)$, while the other choices rearrange or change signs of the coefficients.