Vector Algebra: Key Concepts and Exam Applications: 36 flashcards
All 36 flashcards from this kit, free and with no sign-up.
1. Which vector operation returns a scalar?
Show the answer
The dot product (scalar product) returns a scalar; addition, subtraction, and cross product return vectors.
2. State the formula for the dot product in terms of angle θ.
Show the answer
$\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta$
3. State the formula for the cross product in terms of angle θ.
Show the answer
$\vec{a}\times\vec{b}=|\vec{a}||\vec{b}|\sin\theta\,\hat{n}$, where $\hat{n}$ is the unit vector perpendicular to both.
4. How do you compute $\vec{a}\cdot\vec{b}$ from components?
Show the answer
If $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$, then $\vec{a}\cdot\vec{b}=a_1b_1+a_2b_2+a_3b_3$.
5. What condition must two nonzero vectors satisfy to be perpendicular?
Show the answer
$\vec{a}\perp\vec{b}\iff\vec{a}\cdot\vec{b}=0$.
6. What condition must two nonzero vectors satisfy to be parallel?
Show the answer
$\vec{a}\parallel\vec{b}\iff\vec{a}\times\vec{b}=\vec{0}$.
7. How do you find the angle θ between two vectors?
Show the answer
$\cos\theta=\dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}$, then take $\cos^{-1}$.
8. If $|\vec{a}\cdot\vec{b}|=|\vec{a}\times\vec{b}|$, what is the acute angle between $\vec{a}$ and $\vec{b}$?
Show the answer
$\theta=\pi/4$ (or $45^\circ$).
9. What is the projection of $\vec{a}$ on $\vec{b}$?
Show the answer
$\dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|}$ (a scalar).
10. What is the unit vector in the direction of $\vec{a}$?
Show the answer
$\hat{a}=\dfrac{\vec{a}}{|\vec{a}|}$.
11. How do you get a vector of magnitude $k$ in the direction of $\vec{a}$?
Show the answer
$k\dfrac{\vec{a}}{|\vec{a}|}$.
12. What is the unit vector perpendicular to both $\vec{a}$ and $\vec{b}$?
Show the answer
$\pm\dfrac{\vec{a}\times\vec{b}}{|\vec{a}\times\vec{b}|}$ (two possible directions).
13. What is the area of a parallelogram with adjacent sides $\vec{a}$ and $\vec{b}$?
Show the answer
$|\vec{a}\times\vec{b}|$.
14. What is the area of a triangle with adjacent sides $\vec{a}$ and $\vec{b}$?
Show the answer
$\tfrac12|\vec{a}\times\vec{b}|$.
15. How do you test whether points $A$, $B$, $C$ are collinear?
Show the answer
Check whether $\vec{AB}\times\vec{AC}=\vec{0}$ (or $\vec{AB}=\lambda\vec{AC}$).
16. What is the scalar triple product $[\vec{a}\ \vec{b}\ \vec{c}]$?
Show the answer
$\vec{a}\cdot(\vec{b}\times\vec{c})$.
17. What condition must three vectors satisfy to be coplanar?
Show the answer
$[\vec{a}\ \vec{b}\ \vec{c}]=0$.
18. What is the volume of a parallelepiped with edges $\vec{a},\vec{b},\vec{c}$?
Show the answer
$|[\vec{a}\ \vec{b}\ \vec{c}]|$.
19. What are direction ratios and direction cosines of $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$?
Show the answer
Direction ratios: $a_1,a_2,a_3$. Direction cosines: $\dfrac{a_1}{|\vec{a}|},\dfrac{a_2}{|\vec{a}|},\dfrac{a_3}{|\vec{a}|}$, satisfying $l^2+m^2+n^2=1$.
20. How is $\vec{AB}$ expressed in terms of position vectors?
Show the answer
$\vec{AB}=\vec{OB}-\vec{OA}$.
21. Is the cross product commutative?
Show the answer
No, it is anti-commutative: $\vec{a}\times\vec{b}=-(\vec{b}\times\vec{a})$.
22. Simplify $(\vec{a}-\vec{b})\times(\vec{a}+\vec{b})$.
Show the answer
$2(\vec{a}\times\vec{b})$.
23. Simplify $(\vec{a}+\vec{b})\cdot(\vec{a}-\vec{b})$.
Show the answer
$|\vec{a}|^2-|\vec{b}|^2$.
24. If $\vec{a},\vec{b},\vec{c}$ are unit vectors with $\vec{a}+\vec{b}+\vec{c}=\vec{0}$, what is $\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a}$?
Show the answer
$-3/2$.
25. State Lagrange's identity for $|\vec{a}\times\vec{b}|^2$.
Show the answer
$|\vec{a}\times\vec{b}|^2=|\vec{a}|^2|\vec{b}|^2-(\vec{a}\cdot\vec{b})^2$.
26. What is the value of $\hat{i}\cdot(\hat{j}\times\hat{k})$?
Show the answer
$1$.
27. If $\vec{a},\vec{b},\vec{c}$ are coplanar, are $\vec{a}+\vec{b}$, $\vec{b}+\vec{c}$, $\vec{c}+\vec{a}$ also coplanar?
Show the answer
Yes, they are coplanar.
28. In parallelogram $ABCD$, express $\vec{AC}$ and $\vec{DB}$ in terms of $\vec{AB}$ and $\vec{AD}$.
Show the answer
$\vec{AC}=\vec{AB}+\vec{AD}$ and $\vec{DB}=\vec{AB}-\vec{AD}$.
29. If the diagonals of a parallelogram are equal in magnitude, what type of parallelogram is it?
Show the answer
A rectangle.
30. For fixed magnitudes, when is $\vec{a}\cdot\vec{b}$ maximum?
Show the answer
When $\theta=0$ (vectors point in the same direction).
31. If $\vec{a}$ is perpendicular to $\vec{c}$, what is the projection of $\vec{a}$ on $\vec{c}$?
Show the answer
$0$.
32. What is the projection of $\vec{a}$ on itself?
Show the answer
$|\vec{a}|$.
33. Given unit vector $\vec{b}$ and $(2\vec{x}-2\vec{b})\cdot(\vec{x}+\vec{b})=30$, find $|\vec{x}|$.
Show the answer
$|\vec{x}|=4$.
34. If $|\vec{a}|=3$, $|\vec{b}|=4$, $|\vec{c}|=4$ and each is perpendicular to the sum of the other two, find $|\vec{a}+\vec{b}+\vec{c}|$.
Show the answer
$\sqrt{41}$.
35. Is $\vec{a}\times\vec{b}$ perpendicular to both $\vec{a}$ and $\vec{b}$?
Show the answer
Yes, by definition of the cross product.
36. What is the value of $\vec{a}\cdot(\vec{b}\times\vec{c})$ under cyclic permutation of $\vec{a},\vec{b},\vec{c}$?
Show the answer
It remains unchanged.
Your notes can be a kit like these
Every kit here was made from real study material. Point Veda at a chapter, a PDF, a photo or a link and get notes, flashcards, practice MCQs, games and a scored test — free, in 42 languages.
- Chapter notes
- Flashcards
- Practice MCQs
- Scored tests
- Exam prep
Get Veda free See what students get
Free to start · No ads · Works in 42 languages