Sections 1 to 7: 3-D coordinates, vectors, dot product, cross product, lines, planes, quadric surfaces第 1 至 7 节：三维坐标、向量、点积、叉积、直线、平面、二次曲面CALC III
Name:姓名：Date:日期：
PART I · CORE TECHNIQUESComputational fluency · 28 marks计算流畅度 · 28分
Vectors, the Dot Product, and the Cross Product向量、点积与叉积
Show all component-level working. Express magnitudes in exact simplified radical form. State which formula you are using at each key step (dot product, cross product, etc.).展示所有分量级别的计算过程。用精确化简的根式形式表示模长。在每个关键步骤说明所用公式（点积、叉积等）。
Q1MEDIUMCOREvectors: components, magnitude, unit vector向量：分量、模长、单位向量[8 marks]
(a)Compute $|\mathbf{u}|$ and find the unit vector $\hat{\mathbf{u}}$ in the direction of $\mathbf{u}$.计算 $|\mathbf{u}|$，并求 $\mathbf{u}$ 方向上的单位向量 $\hat{\mathbf{u}}$。[3]
(b)Compute the vector $3\mathbf{u} - 2\mathbf{v}$ and find its magnitude.计算向量 $3\mathbf{u} - 2\mathbf{v}$，并求其模长。[3]
(c)A vector $\mathbf{w}$ has initial point $P = (1, -2, 4)$ and terminal point $Q = (3, 0, 1)$. Write $\mathbf{w}$ in component form and verify that $|\mathbf{w}| \ne |\mathbf{u}|$.向量 $\mathbf{w}$ 的起点为 $P = (1, -2, 4)$，终点为 $Q = (3, 0, 1)$。写出 $\mathbf{w}$ 的分量形式，并验证 $|\mathbf{w}| \ne |\mathbf{u}|$。[2]
(b)Find the angle $\theta$ between $\mathbf{a}$ and $\mathbf{b}$. Leave your answer as an exact inverse-cosine expression and state whether the angle is acute, right, or obtuse.求 $\mathbf{a}$ 与 $\mathbf{b}$ 之间的夹角 $\theta$。将答案保留为精确的反余弦表达式，并判断该角是锐角、直角还是钝角。[3]
(c)Compute the scalar projection of $\mathbf{a}$ onto $\mathbf{b}$, and then the vector projection $\operatorname{proj}_{\mathbf{b}} \mathbf{a}$.计算 $\mathbf{a}$ 在 $\mathbf{b}$ 上的标量投影，以及向量投影 $\operatorname{proj}_{\mathbf{b}} \mathbf{a}$。[3]
(d)A constant force $\mathbf{F} = \langle 2, 3, -1 \rangle$ N moves a particle along the straight path from $A = (0,0,0)$ to $B = (4,-1,2)$ (distances in metres). Compute the work done.恒力 $\mathbf{F} = \langle 2, 3, -1 \rangle$ N 将质点沿直线从 $A = (0,0,0)$ 移动到 $B = (4,-1,2)$（距离单位为米）。计算所做的功。[2]
(a)Compute $\mathbf{p} \times \mathbf{q}$ using the $3 \times 3$ determinant formula.用 $3 \times 3$ 行列式公式计算 $\mathbf{p} \times \mathbf{q}$。[4]
(b)Verify, by direct computation, that $\mathbf{p} \times \mathbf{q}$ is orthogonal to both $\mathbf{p}$ and $\mathbf{q}$.通过直接计算，验证 $\mathbf{p} \times \mathbf{q}$ 与 $\mathbf{p}$ 和 $\mathbf{q}$ 均正交。[2]
(c)Find the area of the parallelogram with adjacent sides $\mathbf{p}$ and $\mathbf{q}$, and hence find the area of the triangle with the same two sides.求以 $\mathbf{p}$ 和 $\mathbf{q}$ 为邻边的平行四边形的面积，进而求以这两边为边的三角形面积。[2]
(d)Let $\mathbf{r} = \langle 0, 2, 1 \rangle$. Compute the scalar triple product $\mathbf{p} \cdot (\mathbf{q} \times \mathbf{r})$ and hence find the volume of the parallelepiped with edge vectors $\mathbf{p}$, $\mathbf{q}$, $\mathbf{r}$.设 $\mathbf{r} = \langle 0, 2, 1 \rangle$。计算混合积 $\mathbf{p} \cdot (\mathbf{q} \times \mathbf{r})$，进而求以 $\mathbf{p}$、$\mathbf{q}$、$\mathbf{r}$ 为棱向量的平行六面体的体积。[2]
PART II · DEFINITIONS AND PROOFRigorous arguments · 26 marks严谨论证 · 26分
Orthogonality, Projections, and the Triple Product正交性、投影与混合积
These items are graded on logical completeness. Write each proof as a sequence of equalities or inequalities with explicit reasons. Do not skip steps that rely on definition. A correct geometric argument supplemented by coordinate algebra is acceptable for the proof questions.本部分按逻辑完整性评分。将每个证明写成带有明确理由的等式或不等式序列。不要跳过依赖定义的步骤。对于证明题，辅以坐标代数的正确几何论证亦可接受。
Q4HARDPROOFcross product is orthogonal to both factors叉积与两个因子均正交[8 marks]
Let $\mathbf{u} = \langle u_1, u_2, u_3 \rangle$ and $\mathbf{v} = \langle v_1, v_2, v_3 \rangle$ be vectors in $\mathbb{R}^3$, and let $\mathbf{w} = \mathbf{u} \times \mathbf{v}$.设 $\mathbf{u} = \langle u_1, u_2, u_3 \rangle$，$\mathbf{v} = \langle v_1, v_2, v_3 \rangle$ 为 $\mathbb{R}^3$ 中的向量，令 $\mathbf{w} = \mathbf{u} \times \mathbf{v}$。
(a)Write out $\mathbf{w}$ in full component form using the determinant definition.用行列式定义写出 $\mathbf{w}$ 的完整分量形式。[2]
(b)Prove that $\mathbf{u} \cdot \mathbf{w} = 0$, justifying each algebraic step.证明 $\mathbf{u} \cdot \mathbf{w} = 0$，并对每个代数步骤给出理由。[3]
(c)By an analogous argument, explain (without fully repeating the algebra) why $\mathbf{v} \cdot \mathbf{w} = 0$, and state what geometric conclusion follows about the direction of $\mathbf{u} \times \mathbf{v}$.用类似的论证（无需完全重复代数过程）解释为何 $\mathbf{v} \cdot \mathbf{w} = 0$，并说明由此可得出关于 $\mathbf{u} \times \mathbf{v}$ 方向的几何结论。[3]
Q5HARDPROOFvector projection formula and distance from a point to a line向量投影公式与点到直线距离[10 marks]
Let $\ell$ be the line in $\mathbb{R}^3$ passing through point $A$ with direction vector $\mathbf{d}$ (where $\mathbf{d} \ne \mathbf{0}$), and let $P$ be a point not on $\ell$. Write $\overrightarrow{AP} = \mathbf{v}$.设 $\ell$ 为 $\mathbb{R}^3$ 中过点 $A$、方向向量为 $\mathbf{d}$（$\mathbf{d} \ne \mathbf{0}$）的直线，$P$ 为不在 $\ell$ 上的点，记 $\overrightarrow{AP} = \mathbf{v}$。
(a)The foot of the perpendicular from $P$ to $\ell$ is the point $F$ on $\ell$ minimising $|PF|$. By decomposing $\mathbf{v}$ into its component along $\mathbf{d}$ and its component orthogonal to $\mathbf{d}$, prove that the distance from $P$ to $\ell$ is从 $P$ 到 $\ell$ 的垂足是 $\ell$ 上使 $|PF|$ 最小的点 $F$。将 $\mathbf{v}$ 分解为沿 $\mathbf{d}$ 方向的分量与垂直于 $\mathbf{d}$ 的分量，证明 $P$ 到 $\ell$ 的距离为
$$ d(P,\ell) = \frac{|\mathbf{v} \times \mathbf{d}|}{|\mathbf{d}|}. $$
[5]
(b)Apply the formula to find the distance from the point $P = (2, 1, -1)$ to the line through $A = (0, 0, 0)$ with direction $\mathbf{d} = \langle 1, 2, 2 \rangle$.应用该公式，求点 $P = (2, 1, -1)$ 到过 $A = (0, 0, 0)$、方向为 $\mathbf{d} = \langle 1, 2, 2 \rangle$ 的直线的距离。[3]
(c)Confirm your answer to (b) by independently computing the vector $\overrightarrow{FP}$ (the perpendicular component) and finding its magnitude.通过独立计算向量 $\overrightarrow{FP}$（垂直分量）并求其模长，验证 (b) 的答案。[2]
Q6HARDPROOFscalar triple product volume formula and coplanarity混合积体积公式与共面性[8 marks]
Let $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ be vectors in $\mathbb{R}^3$ forming the three edge-vectors of a parallelepiped from a common vertex.设 $\mathbf{a}$、$\mathbf{b}$、$\mathbf{c}$ 为 $\mathbb{R}^3$ 中的向量，构成平行六面体从公共顶点出发的三条棱向量。
(a)The base of the parallelepiped is the parallelogram spanned by $\mathbf{b}$ and $\mathbf{c}$. Its area is $|\mathbf{b} \times \mathbf{c}|$. The height is the absolute value of the component of $\mathbf{a}$ in the direction of $\mathbf{b} \times \mathbf{c}$. Using these facts, prove that the volume of the parallelepiped equals $|{\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})}|$.平行六面体的底面是由 $\mathbf{b}$ 和 $\mathbf{c}$ 张成的平行四边形，面积为 $|\mathbf{b} \times \mathbf{c}|$。高为 $\mathbf{a}$ 在 $\mathbf{b} \times \mathbf{c}$ 方向上的分量的绝对值。利用这些事实，证明平行六面体的体积等于 $|{\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})}|$。[4]
(b)State and prove the criterion: three vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ are coplanar if and only if $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0$.陈述并证明以下判定准则：三个向量 $\mathbf{a}$、$\mathbf{b}$、$\mathbf{c}$ 共面，当且仅当 $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0$。[2]
PART III · APPLICATIONS AND SYNTHESISExtended problems · 28 marks综合应用题 · 28分
Lines, Planes, Distance, and Quadric Surfaces直线、平面、距离与二次曲面
Set up each problem cleanly. For lines and planes, state the formula you are using (vector, parametric, or scalar form) before substituting values. Check final answers for geometric consistency.整洁地建立每道题的框架。对于直线和平面，在代入数值前说明所用公式（向量式、参数式或标量式）。检查最终答案的几何一致性。
Q7MEDIUMAPPLIEDlines in space: vector, parametric, symmetric equations空间直线：向量式、参数式、对称式方程[8 marks]
Consider the line $\ell$ passing through $P_0 = (1, -2, 3)$ with direction vector $\mathbf{d} = \langle 2, 1, -1 \rangle$.考虑过点 $P_0 = (1, -2, 3)$、方向向量为 $\mathbf{d} = \langle 2, 1, -1 \rangle$ 的直线 $\ell$。
(a)Write the vector equation and the parametric equations of $\ell$.写出直线 $\ell$ 的向量方程和参数方程。[2]
(b)Write the symmetric equations of $\ell$.写出直线 $\ell$ 的对称式方程。[1]
(c)Determine whether the point $Q = (5, 0, 1)$ lies on $\ell$. Justify your answer by solving the parametric equations simultaneously.判断点 $Q = (5, 0, 1)$ 是否在直线 $\ell$ 上。通过联立参数方程给出理由。[3]
(d)Find the line $m$ through the point $R = (0, 1, -1)$ that is parallel to $\ell$. Write $m$ in parametric form and confirm that $m$ and $\ell$ are distinct (i.e. not the same line).求过点 $R = (0, 1, -1)$ 且与 $\ell$ 平行的直线 $m$。用参数形式写出 $m$，并确认 $m$ 与 $\ell$ 是不同的直线。[2]
Q8HARDAPPLIEDplanes: equation from three points, distance from a point平面：由三点确定方程、点到平面距离[8 marks]
(a)Find two vectors lying in $\Pi$ and compute their cross product to obtain a normal vector $\mathbf{n}$ to $\Pi$.求平面 $\Pi$ 内的两个向量，并计算其叉积以得到 $\Pi$ 的法向量 $\mathbf{n}$。[3]
(b)Write the scalar equation of $\Pi$ in the form $ax + by + cz = d$.写出平面 $\Pi$ 的标量方程，形式为 $ax + by + cz = d$。[2]
(c)Find the distance from the origin $O = (0,0,0)$ to the plane $\Pi$ using the point-to-plane distance formula. Leave the answer in exact form.用点到平面距离公式求原点 $O = (0,0,0)$ 到平面 $\Pi$ 的距离。答案保留精确形式。[2]
(d)A second plane $\Pi'$ has equation $6x + 3y + 2z = 0$. Determine whether $\Pi$ and $\Pi'$ are parallel, perpendicular, or neither. Justify using the dot product of their normals.第二个平面 $\Pi'$ 的方程为 $6x + 3y + 2z = 0$。判断 $\Pi$ 与 $\Pi'$ 是平行、垂直还是既不平行也不垂直，并用两法向量的点积给出理由。[1]
Q9HARDAPPLIEDdistance between skew lines; line-plane intersection; intersection of two planes异面直线距离；直线与平面交点；两平面的交线[6 marks]
Answer all three parts. Each is self-contained.回答全部三个小题。每小题相互独立。
(a)Lines $L_1$ and $L_2$ are given by直线 $L_1$ 和 $L_2$ 分别为
$$ L_1: \; \mathbf{r}_1 = \langle 1,0,0 \rangle + t\langle 1,1,0 \rangle, \quad L_2: \; \mathbf{r}_2 = \langle 0,1,0 \rangle + s\langle 0,1,1 \rangle. $$
Show that $L_1$ and $L_2$ are skew, and find the distance between them.证明 $L_1$ 与 $L_2$ 是异面直线，并求它们之间的距离。[4]
(b)The line $\mathbf{r} = \langle 2, 0, -1 \rangle + t\langle 1, -1, 3 \rangle$ meets the plane $x + 2y + z = 7$. Find the point of intersection.直线 $\mathbf{r} = \langle 2, 0, -1 \rangle + t\langle 1, -1, 3 \rangle$ 与平面 $x + 2y + z = 7$ 相交。求交点坐标。[2]
Q10HARDAPPLIEDtorque via cross product; intersection line of two planes; quadric surface classification叉积求力矩；两平面交线；二次曲面分类[6 marks]
Answer all three parts. Each is self-contained.回答全部三个小题。每小题相互独立。
(a)A wrench applies a force $\mathbf{F} = \langle 0, 4, -2 \rangle$ N at the point whose position vector from the pivot is $\mathbf{r} = \langle 2, -1, 3 \rangle$ m. Compute the torque vector $\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}$ and find its magnitude, interpreting the direction geometrically.扳手在位置向量（从转轴量起）为 $\mathbf{r} = \langle 2, -1, 3 \rangle$ m 的点处施加力 $\mathbf{F} = \langle 0, 4, -2 \rangle$ N。计算力矩向量 $\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}$，求其模长，并从几何上解释其方向。[2]
(b)Find the parametric equations of the line of intersection of the planes $x + y + z = 6$ and $2x - y + z = 3$. (Hint: the direction of the intersection line is $\mathbf{n}_1 \times \mathbf{n}_2$.)求平面 $x + y + z = 6$ 与 $2x - y + z = 3$ 交线的参数方程。（提示：交线方向为 $\mathbf{n}_1 \times \mathbf{n}_2$。）[2]
(c)Identify the type of quadric surface for each equation, naming the surface and, where relevant, the axis of symmetry.判断下列各方程对应的二次曲面类型，说明曲面名称，并在适用时给出对称轴。