Optimization of a Vector Expression

The question asks to find a maximum, minimum, or extremal value (e.g., farthest point, minimum distance) involving a parameterized vector expression.

csat-suneung 2013 Q25 View
In an equilateral triangle ABC with side length 2, let H be the foot of the perpendicular from vertex A to side BC. When point P moves on line segment AH, find the maximum value of $| \overrightarrow { \mathrm { PA } } \cdot \overrightarrow { \mathrm { PB } } |$, which is $\frac { q } { p }$. Find the value of $p + q$. (Given that $p$ and $q$ are coprime natural numbers.)
csat-suneung 2024 Q30_geometry 4 marks View
In the coordinate plane, there is an equilateral triangle ABC with side length 4. Let D be the point that divides segment AB in the ratio $1:3$, E be the point that divides segment BC in the ratio $1:3$, and F be the point that divides segment CA in the ratio $1:3$. Four points $\mathrm{P}$, $\mathrm{Q}$, $\mathrm{R}$, and $\mathrm{X}$ satisfy the following conditions. (가) $|\overrightarrow{\mathrm{DP}}| = |\overrightarrow{\mathrm{EQ}}| = |\overrightarrow{\mathrm{FR}}| = 1$ (나) $\overrightarrow{\mathrm{AX}} = \overrightarrow{\mathrm{PB}} + \overrightarrow{\mathrm{QC}} + \overrightarrow{\mathrm{RA}}$ When $|\overrightarrow{\mathrm{AX}}|$ is maximized, let the area of triangle PQR be $S$. Find the value of $16S^2$. [4 points]
csat-suneung 2025 Q30G 4 marks View
In the coordinate plane, there is a square ABCD with side length 4. $$|\overrightarrow{\mathrm{XB}} + \overrightarrow{\mathrm{XC}}| = |\overrightarrow{\mathrm{XB}} - \overrightarrow{\mathrm{XC}}|$$ Let $S$ be the figure formed by points X satisfying this condition. For a point P on figure $S$, $$4\overrightarrow{\mathrm{PQ}} = \overrightarrow{\mathrm{PB}} + 2\overrightarrow{\mathrm{PD}}$$ Let Q be the point satisfying this condition. If the maximum and minimum values of $\overrightarrow{\mathrm{AC}} \cdot \overrightarrow{\mathrm{AQ}}$ are $M$ and $m$ respectively, find the value of $M \times m$. [4 points]
gaokao 2015 Q9 View
9. Given $\overrightarrow { A B } \perp \overrightarrow { A C }$, $| \overrightarrow { A B } | = \frac { 1 } { t }$, $| \overrightarrow { A C } | = t$, if point $P$ is a point in the plane of $\triangle A B C$, and $\overrightarrow { A P } = \frac { \overrightarrow { A B } } { | \overrightarrow { A B } | } + \frac { \overrightarrow { A C } } { | \overrightarrow { A C } | }$, then the maximum value of $\overrightarrow { P B } \cdot \overrightarrow { P C }$ equals
A. 13
B. 15
C. 19
D. 21
gaokao 2015 Q9 View
9. Points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ move on the circle $\chi ^ { 2 } + y ^ { 2 } = 1$, and $\mathrm { AB } \perp \mathrm { BC }$. If point P has coordinates $( 2,0 )$, then the maximum value of $| \overrightarrow { P A } + \overrightarrow { P B } + \overrightarrow { P C } |$ is
A. 6
B. 7
C. 8
D. 9
gaokao 2015 Q14 5 marks View
In isosceles trapezoid ABCD, $\mathrm{AB} \parallel \mathrm{DC}$, $\mathrm{AB} = 2$, $\mathrm{BC} = 1$, $\angle \mathrm{ABC} = 60°$. Moving points E and F are on segments BC and DC respectively, with $\overrightarrow{\mathrm{BE}} = \lambda\overrightarrow{\mathrm{BC}}$, $\overrightarrow{\mathrm{DF}} = \frac{1}{9\lambda}\overrightarrow{\mathrm{DC}}$. Then the minimum value of $\overrightarrow{\mathrm{AE}} \cdot \overrightarrow{\mathrm{AF}}$ is .
gaokao 2017 Q12 View
12. Given that $\triangle A B C$ is an equilateral triangle with side length 2, and $P$ is a point in the plane $A B C$, the minimum value of $\overrightarrow { P A } \cdot ( \overrightarrow { P B } + \overrightarrow { P C } )$ is
A. $- 2$
B. $- \frac { 3 } { 2 }$
C. $- \frac { 4 } { 3 }$
D. $- 1$
II. Fill in the Blank: This section has 4 questions, each worth 5 points.
jee-advanced 2008 Q7 View
Let two non-collinear unit vectors $\hat { a }$ and $\hat { b }$ form an acute angle. A point $P$ moves so that at any time $t$ the position vector $\overrightarrow { O P }$ (where $O$ is the origin) is given by $\hat { a } \cos t + \hat { b } \sin t$. When $P$ is farthest from origin $O$, let $M$ be the length of $\overrightarrow { O P }$ and $\hat { u }$ be the unit vector along $\overrightarrow { O P }$. Then,
(A) $\hat { u } = \frac { \hat { a } + \hat { b } } { | \hat { a } + \hat { b } | }$ and $M = ( 1 + \hat { a } \cdot \hat { b } ) ^ { \frac { 1 } { 2 } }$
(B) $\hat { u } = \frac { \hat { a } - \hat { b } } { | \hat { a } - \hat { b } | }$ and $M = ( 1 + \hat { a } \cdot \hat { b } ) ^ { \frac { 1 } { 2 } }$
(C) $\hat { u } = \frac { \hat { a } + \hat { b } } { | \hat { a } + \hat { b } | }$ and $M = ( 1 + 2 \hat { a } \cdot \hat { b } ) ^ { \frac { 1 } { 2 } }$
(D) $\hat { u } = \frac { \hat { a } - \hat { b } } { | \hat { a } - \hat { b } | }$ and $M = ( 1 + 2 \hat { a } \cdot \hat { b } ) ^ { \frac { 1 } { 2 } }$
jee-advanced 2019 Q14 View
Let $\vec{a} = 2\hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + \hat{k}$ be two vectors. Consider a vector $\vec{c} = \alpha\vec{a} + \beta\vec{b}$, $\alpha, \beta \in \mathbb{R}$. If the projection of $\vec{c}$ on the vector $(\vec{a} + \vec{b})$ is $3\sqrt{2}$, then the minimum value of $(\vec{c} - (\vec{a} \times \vec{b})) \cdot \vec{c}$ equals
jee-main 2019 Q87 View
Let $\vec { a } = 3 \hat { i } + 2 \hat { j } + x \hat { k }$ and $\vec { b } = \hat { i } - \hat { j } + \hat { k }$, for some real $x$. Then the condition for $| \vec { a } \times \vec { b } | = r$ to follow
(1) $0 < r \leq \sqrt { \frac { 3 } { 2 } }$
(2) $r \geq 5 \sqrt { \frac { 3 } { 2 } }$
(3) $\sqrt { \frac { 3 } { 2 } } < r \leq 3 \sqrt { \frac { 3 } { 2 } }$
(4) $3 \sqrt { \frac { 3 } { 2 } } < r < 5 \sqrt { \frac { 3 } { 2 } }$
jee-main 2023 Q89 View
Let $\vec{v} = \alpha\hat{i} + 2\hat{j} - 3\hat{k}$, $\vec{w} = 2\alpha\hat{i} + \hat{j} - \hat{k}$, and $\vec{u}$ be a vector such that $|\vec{u}| = \alpha > 0$. If the minimum value of the scalar triple product $[\vec{u}\, \vec{v}\, \vec{w}]$ is $-\alpha\sqrt{3401}$, and $|\vec{u} \cdot \hat{i}|^2 = \frac{m}{n}$ where $m$ and $n$ are coprime natural numbers, then $m + n$ is equal to $\_\_\_\_$.
jee-main 2024 Q77 View
Consider three vectors $\vec { a } , \vec { b } , \vec { c }$. Let $| \vec { a } | = 2 , | \vec { b } | = 3$ and $\vec { a } = \vec { b } \times \vec { c }$. If $\alpha \in \left[ 0 , \frac { \pi } { 3 } \right]$ is the angle between the vectors $\vec { b }$ and $\vec { c }$, then the minimum value of $27 | \vec { c } - \vec { a } | ^ { 2 }$ is equal to:
(1) 110
(2) 124
(3) 121
(4) 105