Core Pure 1 Chapter 9: Vectors

21 flashcards · Mathematics · by TheoTyson

1. What is the vector equation of a straight line passing through the position vector aaa , parallel to the vector bbb ?

   r=a+λbr = a + \lambda br=a+λb (where λ\lambdaλ is scalar)

2. What is the vector equation of a straight line passing through the position vectors ccc and ddd ?

   r=c+λr = c + \lambdar=c+λ (d−c)(d - c)(d−c) (where λ\lambdaλ is scalar)

3. What is the cartesian form of the equation r=r=r= aaa + λ+ \, \lambda+λ bbb ?

   x − a1b1=y − a2b2=z − a3b3\frac{x \, - \, a_1}{b_1} = \frac{y \, - \, a_2}{b_2} = \frac{z \, - \, a_3}{b_3}b1​x−a1​​=b2​y−a2​​=b3​z−a3​​

4. What is the vector equation of a plane passing through the position vector aaa , and bbb and ccc are non parallel vectors on the plane?

   r=a+λb+μcr = a + \lambda b + \mu cr=a+λb+μc (where λ\lambdaλ and μ\muμ are scalar)

5. What is the cartesian equation of a plane?

   ax+by+cz=dax + by + cz = dax+by+cz=d (where aaa , bbb , ccc , and ddd are constants, and the normal vector is abc\begin{matrix} a \\ b \\ c \end{matrix}abc​ )

6. What is a normal vector?

   Vector that is perpendicular to the plane. n=an = an=a i +++ bbb j +++ ccc k

7. What is the formula for the scalar product of two vectors aaa and bbb using the angle between them?

   a.ba.ba.b === ∣a∣\lvert a \rvert∣a∣ ∣b∣\lvert b \rvert∣b∣ cosθcos \thetacosθ (where θ\thetaθ is the angle between aaa and bbb )

8. What is the angle between two position vectors aaa and bbb ?

   cosθcos \thetacosθ = a.b∣a∣∣b∣\frac{a.b}{\lvert a \rvert \lvert b \rvert }∣a∣∣b∣a.b​

9. What is the scalar product of two perpendicular vectors?

   a.ba.ba.b =0= 0=0

10. What is the scalar product of two parallel vectors?

    a.ba.ba.b =∣a∣∣b∣= \lvert a \rvert \lvert b \rvert=∣a∣∣b∣

11. How do you work out the scalar product of two vectors in cartesian form?

    a.ba.ba.b === a1b1a_1 b_1a1​b1​ +++ a2b2a_2 b_2a2​b2​ +++ a3b3a_3 b_3a3​b3​

12. What is the acute angle between two intersecting straight lines? (Angle between two lines)

    cosθ=cos \theta =cosθ= ∣b1.b2∣b1∣∣b2∣∣\lvert \frac{b_1 . b_2}{\lvert b_1 \rvert \lvert b_2 \rvert} \rvert∣∣b1​∣∣b2​∣b1​.b2​​∣

13. What is the scalar product form of the equation of a plane?

    r.nr.nr.n =k=k=k (where k=a.nk = a.nk=a.n for any point with position vector aaa , and nnn is the normal of the plane)

14. What is the acute angle between the line with the equation r=a+λbr = a + \lambda br=a+λb and the plane with the equation r.n=kr.n = kr.n=k ? (Angle between line and plane)

    sinθsin \thetasinθ =∣b.n∣b∣∣n∣∣= \lvert \frac{b.n}{\lvert b \rvert \lvert n \rvert} \rvert=∣∣b∣∣n∣b.n​∣

15. What is the acute angle between the plane with the equation r.n1=kr.n_1 = kr.n1​=k and the plane with the equation r.n2=kr.n_2 = kr.n2​=k ? (Angle between two planes)

    cosθcos \thetacosθ =∣n1.n2∣n1∣∣n2∣∣= \lvert \frac{n_1.n_2}{\lvert n_1 \rvert \lvert n_2 \rvert} \rvert=∣∣n1​∣∣n2​∣n1​.n2​​∣

16. What are skew lines?

    Two straight lines that are not parallel and do not intersect.

17. What is the shortest distance between two non intersecting lines?

    Line segment ABABAB such that AAA lies on l1l_1l1​ , and BBB lies on l2l_2l2​ , and ABABAB is perpendicular to both lines.

18. What is the shortest distance between a point and a line?

    A line through PPP which is perpendicular to lll .

19. What is the shortest distance between a point and a plane?

    A line through PPP which is parallel to the normal vector of the plane, nnn .

20. What is the formula for kkk (the length of the perpendicular from the origin to a plane Π\PiΠ )?

    r.n^r.\hat{n}r.n^ === kkk (where n^\hat{n}n^ is a unit vector perpendicular to Π\PiΠ ).

21. What is the perpendicular distance between the point ( α\alphaα ,,, β,\beta ,β, γ\gammaγ ) and the plane with the equation ax + by + cz=dax \, + \, by \, + \, cz = dax+by+cz=d ?

    aα + bβ + cγ − da2 + b2 + c2\frac{a \alpha \, + \, b \beta \, + \, c \gamma \, - \, d}{\sqrt{a^2 \, + \, b^2 \, + \, c^2}}a2+b2+c2 ​aα+bβ+cγ−d​