Vectors for Geometric Reasoning - Math 3

1. Vector $\vec{a} = (3, 4)$ represents a displacement from point $M$ to point $N$. Vector $\vec{b} = (-2, 1)$ represents a displacement from point $N$ to point $P$. If a fourth point $Q$ is positioned such that $MNPQ$ forms a parallelogram, what is the displacement vector from $M$ to $Q$?
   Answer: $(-2, 1)$ In parallelogram $MNPQ$, opposite sides are parallel and equal. Since $\overrightarrow{MN} = (3, 4)$ and $\overrightarrow{NP} = (-2, 1)$, we need $\overrightarrow{QP} = \overrightarrow{MN} = (3, 4)$ and $\overrightarrow{MQ} = \overrightarrow{NP} = (-2, 1)$. Therefore, the displacement from $M$ to $Q$ is $(-2, 1)$. Choice A represents $\overrightarrow{MP}$. Choice B is incorrect vector addition. Choice D represents the opposite of the correct answer.
2. In regular hexagon $ABCDEF$ with center $O$, let $\overrightarrow{OA} = \vec{u}$ and $\overrightarrow{OB} = \vec{v}$. Express $\overrightarrow{AD}$ in terms of $\vec{u}$ and $\vec{v}$, where $D$ is the vertex opposite to $A$.
   Answer: $-2\vec{u}$ In a regular hexagon, vertex $D$ is diametrically opposite to vertex $A$, so $\overrightarrow{OD} = -\overrightarrow{OA} = -\vec{u}$. Therefore, $\overrightarrow{AD} = \overrightarrow{OD} - \overrightarrow{OA} = -\vec{u} - \vec{u} = -2\vec{u}$. Choice A incorrectly includes $\vec{v}$. Choice C represents a vector between adjacent vertices. Choice D represents the vector $\overrightarrow{AB}$ in the regular hexagon.
3. Vector $\vec{a} = (2, -3)$ represents a translation. After applying this translation to point $T(1, 4)$, the resulting point is reflected across the line $y = x$. What are the coordinates of the final point?
   Answer: $(1, 3)$ This question tests your ability to perform geometric transformations in sequence—first a translation, then a reflection. When you see multiple transformations, always apply them one at a time in the given order. Start by applying the translation vector $\vec{a} = (2, -3)$ to point $T(1, 4)$. A translation adds the vector components to the original coordinates: $(1, 4) + (2, -3) = (1+2, 4+(-3)) = (3, 1)$. This gives you the intermediate point after translation. Next, reflect this intermediate point $(3, 1)$ across the line $y = x$. Reflection across $y = x$ has a simple rule: swap the $x$ and $y$ coordinates. So $(3, 1)$ becomes $(1, 3)$, which is answer C. Let's examine why the other answers are wrong. Answer A $(1, -3)$ appears to apply only the translation incorrectly, possibly subtracting instead of adding the $y$-component. Answer B $(3, 1)$ is the intermediate point after translation but before reflection—a common error when students forget to complete the second transformation. Answer D $(-3, 1)$ seems to confuse the order of operations or misapply the translation vector. Remember this key strategy: for multi-step transformation problems, work systematically through each transformation in order and write down your intermediate results. Don't try to combine steps mentally, as this often leads to sign errors or skipped operations. Also memorize that reflection across $y = x$ always means swapping coordinates.
4. In parallelogram $PQRS$, the diagonals intersect at point $T$. If $\overrightarrow{PQ} = \vec{u}$ and $\overrightarrow{PS} = \vec{v}$, which expression correctly represents $\overrightarrow{TR}$?
   Answer: $\frac{1}{2}(\vec{u} + \vec{v})$ In a parallelogram, the diagonals bisect each other. Point $T$ is the midpoint of both diagonals. $\overrightarrow{PR} = \overrightarrow{PQ} + \overrightarrow{QR} = \overrightarrow{PQ} + \overrightarrow{PS} = \vec{u} + \vec{v}$. Since $T$ is the midpoint of diagonal $PR$, we have $\overrightarrow{PT} = \frac{1}{2}\overrightarrow{PR} = \frac{1}{2}(\vec{u} + \vec{v})$. Therefore, $\overrightarrow{TR} = \overrightarrow{PR} - \overrightarrow{PT} = (\vec{u} + \vec{v}) - \frac{1}{2}(\vec{u} + \vec{v}) = \frac{1}{2}(\vec{u} + \vec{v})$. Choice B represents half a side vector. Choice C represents the negative of choice B. Choice D represents the vector from $T$ to $P$.
5. In triangle $DEF$, point $G$ lies on side $DE$ such that $DG:GE = 2:1$. Point $H$ lies on side $DF$ such that $DH:HF = 1:2$. If $\overrightarrow{DE} = \vec{p}$ and $\overrightarrow{DF} = \vec{q}$, what is $\overrightarrow{GH}$ in terms of $\vec{p}$ and $\vec{q}$?
   Answer: $\frac{1}{3}\vec{q} - \frac{2}{3}\vec{p}$ Since $DG:GE = 2:1$, point $G$ divides $DE$ in ratio $2:1$, so $\overrightarrow{DG} = \frac{2}{3}\vec{p}$. Since $DH:HF = 1:2$, point $H$ divides $DF$ in ratio $1:2$, so $\overrightarrow{DH} = \frac{1}{3}\vec{q}$. Therefore, $\overrightarrow{GH} = \overrightarrow{DH} - \overrightarrow{DG} = \frac{1}{3}\vec{q} - \frac{2}{3}\vec{p}$. Choice B incorrectly uses the position vectors rather than their difference. Choice C reverses the coefficients. Choice D has an incorrect sign.
6. Vector $\vec{u} = (3, -1)$ represents a displacement. If this displacement is applied twice consecutively, followed by the displacement $\vec{v} = (-2, 5)$, what is the total displacement from the starting position?
   Answer: $(4, 3)$ Applying displacement $\vec{u}$ twice gives $2\vec{u} = 2(3, -1) = (6, -2)$. Then applying $\vec{v}$ gives total displacement $(6, -2) + (-2, 5) = (4, 3)$. Choice A incorrectly subtracts $\vec{v}$ instead of adding it. Choice C uses $\vec{u}$ only once. Choice D adds the magnitudes instead of using vector addition.
7. Triangle $PQR$ has vertices at $P(1, 2)$, $Q(4, 6)$, and $R(7, 1)$. Point $S$ is located such that $\overrightarrow{PS} = \frac{1}{3}\overrightarrow{PQ} + \frac{2}{3}\overrightarrow{PR}$. What type of point is $S$ relative to triangle $PQR$?
   Answer: $S$ lies on the side $QR$ and divides it in ratio $1:2$ Since the coefficients $\frac{1}{3}$ and $\frac{2}{3}$ sum to 1, point $S$ lies on side $QR$. To find the exact position, note that $\overrightarrow{PS} = \frac{1}{3}\overrightarrow{PQ} + \frac{2}{3}\overrightarrow{PR}$ can be rewritten as $S = \frac{2}{3}R + \frac{1}{3}Q$, which means $S$ divides $QR$ in the ratio $QS:SR = 2:1$ (or $S$ divides $QR$ in ratio $1:2$ from $Q$ to $R$). Choice A reverses the ratio. Choice C is incorrect since the centroid would require equal coefficients of $\frac{1}{3}$. Choice D is incorrect since $S$ lies on side $QR$.
8. Point $M$ is the centroid of triangle $ABC$. If $\overrightarrow{MA} = \vec{x}$, $\overrightarrow{MB} = \vec{y}$, and $\overrightarrow{MC} = \vec{z}$, which relationship must hold among these vectors?
   Answer: $\vec{x} + \vec{y} + \vec{z} = \vec{0}$ When you encounter problems involving centroids and vectors, remember that the centroid has a special balancing property that creates a key vector relationship. The centroid M of triangle ABC is the point where the three medians intersect, and it's located exactly one-third of the way from each side to the opposite vertex. This geometric property translates into a fundamental vector relationship: the sum of all vectors from the centroid to the vertices equals the zero vector. To see why $\vec{x} + \vec{y} + \vec{z} = \vec{0}$, consider that the centroid is the "balance point" of the triangle. If you place equal masses at each vertex, the centroid is where the triangle would balance perfectly. This means the vector forces from the centroid to each vertex must cancel out completely, resulting in $\overrightarrow{MA} + \overrightarrow{MB} + \overrightarrow{MC} = \vec{0}$. Answer choice A is incorrect because the dot products of these vectors aren't necessarily equal - they depend on the specific angles and magnitudes between the vectors. Choice B fails because the vectors $\vec{x}$, $\vec{y}$, and $\vec{z}$ point in different directions from M to each vertex, so they can't be equal unless the triangle degenerates to a point. Choice C is wrong because the centroid isn't equidistant from all vertices unless the triangle is equilateral - in most triangles, these distances vary. Remember this key principle: for any centroid of a triangle, the vectors from the centroid to the three vertices always sum to zero. This relationship appears frequently in vector geometry problems.
9. In triangle $DEF$, $\vec{DE} = (4, -2)$ and $\vec{DF} = (1, 6)$. What is the vector from the midpoint of side $DE$ to the midpoint of side $EF$?
   Answer: $(-\frac{3}{2}, 4)$ Let $M$ be the midpoint of $DE$ and $N$ be the midpoint of $EF$. First find $\vec{EF} = \vec{DF} - \vec{DE} = (1, 6) - (4, -2) = (-3, 8)$. The vector from midpoint to midpoint is $\vec{MN} = \frac{1}{2}\vec{EF} = \frac{1}{2}(-3, 8) = (-\frac{3}{2}, 4)$. Choice B uses incorrect y-component. Choice C has the wrong signs. Choice D incorrectly uses the average of the original vectors.
10. In triangle $XYZ$, point $P$ satisfies the equation $3\overrightarrow{PX} + 2\overrightarrow{PY} + \overrightarrow{PZ} = \vec{0}$. What can be concluded about the location of point $P$?
    Answer: $P$ is the weighted centroid with masses 3, 2, 1 at $X$, $Y$, $Z$ respectively The equation $3\overrightarrow{PX} + 2\overrightarrow{PY} + \overrightarrow{PZ} = \vec{0}$ represents a weighted average where point $P$ is the balance point (center of mass) when masses 3, 2, and 1 are placed at vertices $X$, $Y$, and $Z$ respectively. This is the definition of a weighted centroid. Choice A is incorrect because the regular centroid has equal weights (coefficients of 1). Choice B is incorrect as this equation doesn't place $P$ on any side. Choice D is too vague and doesn't capture the specific weighted centroid relationship.
11. Vectors $\overrightarrow{OA} = (2, 3)$ and $\overrightarrow{OB} = (6, 1)$ originate from point $O$. Point $C$ is positioned such that quadrilateral $OACB$ is a parallelogram. If point $D$ is the intersection of diagonals $OC$ and $AB$, what are the coordinates of $\overrightarrow{OD}$?
    Answer: $(4, 2)$ When you encounter parallelogram problems involving vectors, remember that opposite sides are equal and parallel, and diagonals bisect each other. Since OACB is a parallelogram with O as one vertex, we need to find point C. In a parallelogram, opposite sides are equal vectors, so $\overrightarrow{OA} = \overrightarrow{BC}$ and $\overrightarrow{OB} = \overrightarrow{AC}$. This means $\overrightarrow{OC} = \overrightarrow{OA} + \overrightarrow{OB} = (2,3) + (6,1) = (8,4)$. Point D is where diagonals OC and AB intersect. A key property of parallelograms is that diagonals bisect each other, meaning D is the midpoint of both diagonals. Therefore, $\overrightarrow{OD} = \frac{1}{2}\overrightarrow{OC} = \frac{1}{2}(8,4) = (4,2)$. This confirms answer D is correct. Looking at the wrong answers: A) $(3, 1.5)$ appears to be the midpoint of $\overrightarrow{OA}$ and $\overrightarrow{OB}$ averaged incorrectly. B) $(2, 1)$ might come from incorrectly finding the midpoint of diagonal AB instead of using the bisection property. C) $(8, 4)$ is the position vector of point C itself, not the midpoint D. The key strategy here is recognizing that in parallelogram problems, you should immediately think about the diagonal bisection property. When diagonals intersect, that intersection point is always the midpoint of both diagonals. This saves time compared to finding coordinates of all vertices and then calculating intersection points algebraically.
12. Point $A$ is at $(2, 3)$ and point $B$ is at $(5, 7)$. If vector $\vec{AC}$ has the same direction as $\vec{AB}$ but twice the magnitude, what are the coordinates of point $C$?
    Answer: $(8, 11)$ First find $\vec{AB} = (5-2, 7-3) = (3, 4)$. Since $\vec{AC}$ has the same direction but twice the magnitude, $\vec{AC} = 2\vec{AB} = 2(3, 4) = (6, 8)$. Therefore $C = A + \vec{AC} = (2, 3) + (6, 8) = (8, 11)$. Choice B incorrectly adds the doubled vector to point B instead of A. Choice C uses the wrong scalar multiple. Choice D adds the original vector instead of the doubled vector.
13. In quadrilateral $WXYZ$, the midpoint of diagonal $WY$ is $M$ and the midpoint of diagonal $XZ$ is $N$. If $\overrightarrow{WX} = \vec{u}$ and $\overrightarrow{WZ} = \vec{v}$, which condition must be satisfied for $WXYZ$ to be a parallelogram?
    Answer: $\overrightarrow{MN} = \vec{0}$ (the zero vector) In a parallelogram, the diagonals bisect each other, meaning they have the same midpoint. Therefore, $M = N$, which means $\overrightarrow{MN} = \vec{0}$. If the midpoints of the diagonals coincide, then $WXYZ$ is a parallelogram. Choice B represents half the sum of two adjacent side vectors. Choice C represents half the difference of two adjacent side vectors. Choice D represents one-quarter of the sum, which has no geometric significance here.
14. In quadrilateral $KLMN$, $\vec{KL} = (4, 3)$, $\vec{LM} = (-1, 2)$, and $\vec{MN} = (-4, -3)$. What must $\vec{NK}$ be for $KLMN$ to be a closed quadrilateral?
    Answer: $(1, -2)$ When you encounter vector problems involving closed polygons, remember that for any closed figure, the sum of all displacement vectors must equal zero—you end up where you started. To find $\vec{NK}$, you need to use the fact that in a closed quadrilateral, the vector sum around the perimeter equals the zero vector: $\vec{KL} + \vec{LM} + \vec{MN} + \vec{NK} = \vec{0}$. Substituting the given vectors: $(4, 3) + (-1, 2) + (-4, -3) + \vec{NK} = (0, 0)$ Adding the first three vectors component-wise: - x-component: $4 + (-1) + (-4) = -1$ - y-component: $3 + 2 + (-3) = 2$ So $(-1, 2) + \vec{NK} = (0, 0)$, which means $\vec{NK} = (0, 0) - (-1, 2) = (1, -2)$. Looking at the wrong answers: Choice A, $(-1, -2)$, would give a vector sum of $(-2, 0)$, not zero. Choice B, $(-1, 2)$, is actually the negative of what we need—this represents going from K to N instead of N to K. Choice C, $(1, 2)$, has the correct x-component but wrong y-component, resulting in a sum of $(0, 4)$. The correct answer is D: $(1, -2)$. Remember this key principle: for any closed polygon traced by vectors, the sum of all vectors must be zero. This creates a system where once you know all but one vector, the last one is completely determined by the requirement that everything must "close up."
15. In quadrilateral $PQRS$, point $T$ is defined by $\overrightarrow{PT} = \frac{1}{4}\overrightarrow{PQ} + \frac{3}{4}\overrightarrow{PS}$. Point $U$ is defined by $\overrightarrow{PU} = \frac{3}{4}\overrightarrow{PQ} + \frac{1}{4}\overrightarrow{PS}$. If $\overrightarrow{PQ} \perp \overrightarrow{PS}$ and $|\overrightarrow{PQ}| = |\overrightarrow{PS}| = 4$, what is the distance $TU$?
    Answer: $2\sqrt{2}$ When you see vector problems involving linear combinations of perpendicular vectors, set up a coordinate system to make calculations manageable. Since $\overrightarrow{PQ} \perp \overrightarrow{PS}$ and both have magnitude 4, place $P$ at the origin with $\overrightarrow{PQ} = (4,0)$ and $\overrightarrow{PS} = (0,4)$. Now find the positions of points $T$ and $U$: For point $T$: $\overrightarrow{PT} = \frac{1}{4}(4,0) + \frac{3}{4}(0,4) = (1,0) + (0,3) = (1,3)$ For point $U$: $\overrightarrow{PU} = \frac{3}{4}(4,0) + \frac{1}{4}(0,4) = (3,0) + (0,1) = (3,1)$ The distance $TU$ is: $TU = \sqrt{(3-1)^2 + (1-3)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$ Answer choice A ($3\sqrt{2}$) likely comes from incorrectly calculating $\sqrt{(3-1)^2 + (3-1)^2} = \sqrt{8+10} = 3\sqrt{2}$ by mixing up coordinates. Answer choice B ($4$) represents the magnitude of the original vectors, not the distance between the derived points. Answer choice C ($2\sqrt{5}$) suggests calculating $\sqrt{2^2 + 4^2} = 2\sqrt{5}$, perhaps by incorrectly finding the difference vector. The correct answer is D ($2\sqrt{2}$). Strategy tip: When vectors are perpendicular with equal magnitudes, use them as basis vectors in a coordinate system. This transforms abstract vector operations into concrete coordinate calculations, making distance problems much more straightforward.
16. Points $A$, $B$, and $C$ are collinear with $B$ between $A$ and $C$. If $\overrightarrow{AB} = k\overrightarrow{AC}$ where $k$ is a positive constant, and the distance from $A$ to $B$ is 3 units while the distance from $A$ to $C$ is 8 units, what is the value of $k$?
    Answer: $\frac{3}{8}$ Since the points are collinear and $B$ is between $A$ and $C$, vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ point in the same direction. The relationship $\overrightarrow{AB} = k\overrightarrow{AC}$ means $k$ is the ratio of the magnitudes: $k = \frac{|\overrightarrow{AB}|}{|\overrightarrow{AC}|} = \frac{3}{8}$. Choice B gives the reciprocal ratio. Choice C represents $\frac{BC}{AC} = \frac{5}{8}$. Choice D represents $\frac{AB}{BC} = \frac{3}{5}$.

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