文/李潔芸
新東方廣州學校
1 Equation of a Plane 平面方程
· Normal form:
r.n = a.n
其中，n稱之為平面的normal（法向量）。平面Cartesian form中x,y,z的係數，即為法向量的各components。
· Cartesian form:
解題突破口：
px+qy+zr= a.n = k
其中，k由A的坐標決定
Typical Exam Question：
9709_y17_sp_3
Solution：
Step1:Find a direction vector for AB, which is the normal of plane m
Step2:Using the direction vector and the relevant point C to obtain an equation for m
r.n = a.n
Therefore, 2x -2y +z = 4
2 Cross Product Rule 叉乘
non-zero and perpendicular to both of them.
解題突破口：
To memorize:
Typical Exam Question：
Solution：
Step1:Using Product Rule to calculate vector product of relevant vectors, which is the normal for the plane
Step2:Using the direction vector and substitute point (3,-1,2) to obtain an equation for m
r.n = a.n
Therefore, 4x +13y +5z = 9
3Finding the Equation of a Plane
· Given 3 points on a plane三點確定一平面
解題突破口：
比如A(1,2, −1), B(2,1,0), C(−1,3,2)
Step1：Use the equation r.n = a.n
其中，
r is what we want to find,
n is the cross product of 2 vectors parallel to the plane
Step2: use AB and AC to find n
Step3：a = OA, then substitute point A to get a.n
Typical Exam Question：
9709_s16_qp_31
Solution：
Step1：Find AB and AC
Step2: calculate their vector product to find n
Step3： a = OA, then substitute OA to get a.n
Therefore, 5x -2y +3z = 5
· Given a point and a line on the plane: 一點、一直線確定一平面
解題突破口：
Step1: Make 2 points on the line
Step2: Substitute different values for t
Step3: Repeat 3 point process
· Given 2 lines on a plane:
解題突破口：
Step1: Find a point on one line
Step2: Find 2 points on the other line
Step3: Repeat 3 point process
· Determine whether a line lies on a plane判斷直線是否在平面上
解題突破口：
If a line lies on a plane , any two points on the line (e.g. t = 0 and t = 1)should satisfy the plane equation –substitute and see if equation works
Typical Exam Question：
9709_m17_qp_32
Solution：
Step1：Find two points on the line，e.g.λ=0→（1,2,-3）and λ=1→（3,1,-2）
Step2: Substitute their coordinates int the equation of p and verify that both points lies in the plane:
3×1 + 2 - 5×（-3）=20 and 3×3 + 1 - 5×（-2）=20
Therefore, line l lies on plane p.
·Determine whether a line is parallel to plane判斷直線是否和平面平行
解題突破口：
If a line is parallel to plane, the dot product of the direction vector and normal of the plane is zero, or use cross product to find
Typical Exam Question：
9709_s16_qp_32
Solution to (ii)：
Step1: Find vectors of OA and BC, i.e. OA=i+2j+3k, BC= 2i+j-2k
Step2: Use cross product and attempt to calculate vector product of relevant vectors,
（i+2j+3k）×（2i+j-2k）=（-7i+8j-3k）
, which is the normal of the plane p
Step3: Substitute a point to -7x+8y-3k=d to obtain d, e.g. point B
（-7）×0 + 8×4 - 3×1= 29
Therefore, equation of p is -7x+8y-3k = 29
李潔芸老師簡介主授高中物理、Alevel理科、SAT2理科、ACT數學、科學推理，碩士畢業於香港大學。
擁有8年新東方執教經驗，2014年榮獲廣州新東方優秀教師。