Vectors

Vectors give one or two questions: a magnitude, a dot product, a perpendicularity condition or an area from a cross product. Everything follows from component arithmetic in \mathbf{i},\mathbf{j},\mathbf{k}, so the chapter is short and the marks are cheap.

11.1 Vector algebra, position vectors and unit vectors

• \mathbf{a}=a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}, |\mathbf{a}|=\sqrt{a_1^2+a_2^2+a_3^2}. The unit vector is \hat{\mathbf{a}}=\frac{\mathbf{a}}{|\mathbf{a}|} and has magnitude 1.
• \overrightarrow{AB}=\mathbf{b}-\mathbf{a} (head minus tail). The midpoint of A,B has position vector \frac{\mathbf{a}+\mathbf{b}}{2}; the point dividing AB in m:n is \frac{m\mathbf{b}+n\mathbf{a}}{m+n}.
• Add and subtract component by component; k\mathbf{a} scales every component.

(a) W. R. Hamilton, inventor of quaternions

(b) Broom Bridge plaque: i^2 = j^2 = k^2 = ijk = -1

(c) Compass rose: direction and distance

11.2 Dot product, cross product and their applications

• \mathbf{i}\cdot\mathbf{i}=1, \mathbf{i}\cdot\mathbf{j}=0. \cos\theta=\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}. Projection of \mathbf{a} on \mathbf{b} is \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}. Work =\mathbf{F}\cdot\mathbf{d}. |\mathbf{a}+\mathbf{b}|^2=|\mathbf{a}|^2+|\mathbf{b}|^2+2\mathbf{a}\cdot\mathbf{b}.
• \mathbf{i}\times\mathbf{j}=\mathbf{k}, \mathbf{j}\times\mathbf{k}=\mathbf{i}, \mathbf{k}\times\mathbf{i}=\mathbf{j}; reversing the order flips the sign: \mathbf{j}\times\mathbf{i}=-\mathbf{k}. \mathbf{a}\times\mathbf{a}=\mathbf{0}. The result is a vector; the dot product is a scalar.
• Area of parallelogram =|\mathbf{a}\times\mathbf{b}|; of triangle =\frac{1}{2}|\mathbf{a}\times\mathbf{b}|.

11.3 Eight traps in vectors

Vector questions are short, but each mixes a scalar result with a vector result, and the paper loves to test the difference. These eight traps cover the model questions.

1. A vector has magnitude and direction; a scalar has only size. The magnitude of \vec a=(3,4) is \sqrt{3^2+4^2}=5, and the unit vector is \hat a=\frac{\vec a}{|\vec a|}=\left(\frac35,\frac45\right) (Fig 11.1). A unit vector always has length 1, so check this. Do not write |\vec a| as 3+4.
2. Head minus tail. The vector from A to B is \vec{AB}=\vec{OB}-\vec{OA}, the position vector of the end point minus the start. For A(1,2) and B(7,10) this is (6,8). The midpoint has position vector \frac{\vec a+\vec b}{2}, and the internal ratio m:n gives \frac{m\vec b+n\vec a}{m+n}.
3. Parallel means a scalar multiple. \vec a\parallel\vec b when \vec a=\lambda\vec b, so (2,6) and (1,3) are parallel with \lambda=2. Three points are collinear if \vec{AB} and \vec{BC} are parallel. Equal vectors have equal components, not just equal length.
4. The dot product gives a number. \vec a\cdot\vec b=a_1b_1+a_2b_2=|a||b|\cos\theta is a scalar (Fig 11.2). It is zero when the vectors are perpendicular, so (3,4)\cdot(4,-3)=0. Also \vec a\cdot\vec a=|\vec a|^2, \hat i\cdot\hat i=1 and \hat i\cdot\hat j=0.
5. Angle and projection come from the dot product. \cos\theta=\frac{\vec a\cdot\vec b}{|a||b|}, and the scalar projection of \vec a on \vec b is \frac{\vec a\cdot\vec b}{|\vec b|}. The vector projection multiplies that by \hat b, giving \frac{\vec a\cdot\vec b}{|\vec b|^2}\vec b. A question that asks for "the component" wants the scalar one.
6. The cross product gives a vector, and the order matters. \vec a\times\vec b=-\vec b\times\vec a, with magnitude |a||b|\sin\theta, and it is perpendicular to both (Fig 11.3). In the determinant the middle sign flips, which is the term most students get wrong. \vec a\times\vec b=\vec0 means the vectors are parallel.
7. Areas and work come from the two products. The parallelogram on \vec a,\vec b has area |\vec a\times\vec b|, and the triangle has half of that. Work done is W=\vec F\cdot\vec d (a scalar), while the moment of a force is \vec r\times\vec F (a vector). For F=(2,3) and d=(4,1), W=11.
8. Do not treat vectors like numbers. |\vec a+\vec b|\ne|\vec a|+|\vec b| unless the vectors point the same way. (\vec a+\vec b)^2=|a|^2+2\vec a\cdot\vec b+|b|^2, and an expression such as \vec a\cdot\vec b\cdot\vec c has no meaning because the first dot is a scalar. There is no division of one vector by another.

11.3.1 Worked checks

Q. Find the magnitude and unit vector of 3\hat i+4\hat j, and the angle between (3,4) and (2,-1).

Answer. |\vec a|=5 and the unit vector is 0.6\hat i+0.8\hat j. The dot product is 6-4=2, |\vec b|=\sqrt5, so \cos\theta=\frac{2}{5\sqrt5}=0.179 and \theta=79.7^\circ. This is the angle drawn in Fig 11.2.

Q. Find \vec a\times\vec b for \vec a=(1,2,3) and \vec b=(4,5,6), and the area of the parallelogram.

Answer. \vec a\times\vec b=(12-15,\,12-6,\,5-8)=(-3,6,-3). The area is \sqrt{9+36+9}=\sqrt{54}\approx7.35.

Check (-3,6,-3)\cdot(1,2,3)=-3+12-9=0, so the result is perpendicular to \vec a.

Q. Find \lambda so that (2,\lambda) is perpendicular to (3,-4), and the work done by F=(2,3) over d=(4,1).

Answer. Perpendicular means the dot product is zero: 6-4\lambda=0, so \lambda=\frac32. Work =2(4)+3(1)=11 units. Both answers use the scalar dot product.

Practice questions

Set A — Recall (20 questions)

1. The magnitude of 3\mathbf{i}+4\mathbf{j} isa7b1c25d5
2. \mathbf{i}\cdot\mathbf{j} equalsa0b1c\mathbf{k}d-1
3. \mathbf{i}\times\mathbf{j} equalsa-\mathbf{k}b0c\mathbf{k}d1
4. For \mathbf{a}=(1,2,3) and \mathbf{b}=(4,5,6), \mathbf{a}\cdot\mathbf{b} isa28b32c21d15
5. Two non-zero vectors are perpendicular whenatheir dot product is 1btheir dot product is -1ctheir cross product is zerodtheir dot product is zero
6. The unit vector along 3\mathbf{i}+4\mathbf{j} isa3\mathbf{i}+4\mathbf{j}b\frac{3\mathbf{i}+4\mathbf{j}}{25}c\frac{3\mathbf{i}+4\mathbf{j}}{5}d\frac{4\mathbf{i}+3\mathbf{j}}{5}
7. \mathbf{a}\times\mathbf{a} equalsathe zero vectorb|\mathbf{a}|^2c\mathbf{a}d1
8. |\mathbf{a}\times\mathbf{b}| is the area of aatrianglebparallelogramcrectangle onlydcircle
9. The angle between \mathbf{i} and \mathbf{j} isa0°b45°c90°d180°
10. If \mathbf{a} and \mathbf{b} point the same way, \mathbf{a}\cdot\mathbf{b} equalsa0b|\mathbf{a}|+|\mathbf{b}|c\frac{|\mathbf{a}|}{|\mathbf{b}|}d|\mathbf{a}||\mathbf{b}|
11. For A(1,2,3) and B(4,6,8), \overrightarrow{AB} isa3\mathbf{i}+4\mathbf{j}+5\mathbf{k}b5\mathbf{i}+8\mathbf{j}+11\mathbf{k}c-3\mathbf{i}-4\mathbf{j}-5\mathbf{k}d4\mathbf{i}+6\mathbf{j}+8\mathbf{k}
12. The magnitude of \mathbf{i}+\mathbf{j}+\mathbf{k} isa3b\sqrt{3}c1d2
13. \mathbf{j}\times\mathbf{i} equalsa\mathbf{k}b0c-\mathbf{k}d1
14. \mathbf{k}\cdot\mathbf{k} equalsa0b\mathbf{k}c-1d1
15. The projection of \mathbf{a} on \mathbf{b} isa\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}b\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|}c|\mathbf{a}||\mathbf{b}|d\mathbf{a}\times\mathbf{b}
16. Two vectors are parallel whena\mathbf{a}\cdot\mathbf{b}=0b|\mathbf{a}|=|\mathbf{b}|c\mathbf{a}\times\mathbf{b}=\mathbf{0}d\mathbf{a}+\mathbf{b}=\mathbf{0}
17. The position vector of the midpoint of A(2,4) and B(6,8) isa8\mathbf{i}+12\mathbf{j}b4\mathbf{i}+6\mathbf{j}c3\mathbf{i}+5\mathbf{j}d2\mathbf{i}+2\mathbf{j}
18. The magnitude of 2\mathbf{i}-2\mathbf{j}+\mathbf{k} isa3b1c5d9
19. If |\mathbf{a}|=3, |\mathbf{b}|=4 and the angle between them is 90°, then \mathbf{a}\cdot\mathbf{b} isa12b7c1d0
20. The cross product of two vectors is aascalarbvectorcmatrixdplain number

Set B — Understanding (12 questions)

1. The angle between \mathbf{a}=\mathbf{i}+\mathbf{j} and \mathbf{b}=\mathbf{i}-\mathbf{j} isa90°b45°c60°d0°
2. \mathbf{a}=\lambda\mathbf{i}+4\mathbf{j} and \mathbf{b}=2\mathbf{i}-\mathbf{j} are perpendicular when \lambda isa1b4c2d-2
3. The magnitude of \mathbf{i}\times(\mathbf{i}+\mathbf{j}) isa0b2c\sqrt{2}d1
4. The area of the parallelogram with adjacent sides 3\mathbf{i} and 4\mathbf{j} isa7b12c6d1
5. The area of the triangle with sides 2\mathbf{i} and 3\mathbf{j} from a vertex isa3b6c12d5
6. The vector from A(1,1) to B(4,5) has magnitudea7b3c4d5
7. If |\mathbf{a}|=|\mathbf{b}|=1 and \mathbf{a}\cdot\mathbf{b}=\frac{1}{2}, the angle between them isa30°b45°c60°d90°
8. For \mathbf{a}=\mathbf{i}+2\mathbf{j}+3\mathbf{k} and \mathbf{b}=2\mathbf{i}-\mathbf{j}+\mathbf{k}, \mathbf{a}\cdot\mathbf{b} isa5b3c0d7
9. The projection of 3\mathbf{i}+4\mathbf{j} on \mathbf{i} isa5b4c7d3
10. A vector perpendicular to 3\mathbf{i}+4\mathbf{j} isa4\mathbf{i}-3\mathbf{j}b3\mathbf{i}+4\mathbf{j}c4\mathbf{i}+3\mathbf{j}d3\mathbf{i}-4\mathbf{j}
11. If |\mathbf{a}|=2, |\mathbf{b}|=3 and the angle is 30°, then \mathbf{a}\cdot\mathbf{b} isa6b3\sqrt{3}c3d6\sqrt{3}
12. With |\mathbf{a}|=2, |\mathbf{b}|=3 and the angle 30°, |\mathbf{a}\times\mathbf{b}| isa6b3\sqrt{3}c3d1.5

Set C — Application (8 questions)

1. The vectors \mathbf{i}+\lambda\mathbf{j}+\mathbf{k} and 2\mathbf{i}+4\mathbf{j}+2\mathbf{k} are parallel when \lambda isa1b2c4d\frac{1}{2}
2. The magnitude of \mathbf{i}+\mathbf{j} isa2b1c0d\sqrt{2}
3. A force 2\mathbf{i}+3\mathbf{j} moves a body through 4\mathbf{i}+\mathbf{j}. The work done isa11b14c5d6
4. The area of the triangle A(0,0), B(4,0), C(0,3) found by \frac{1}{2}|\overrightarrow{AB}\times\overrightarrow{AC}| isa12b7c6d3.5
5. If \mathbf{a}\perp\mathbf{b} with |\mathbf{a}|=3 and |\mathbf{b}|=4, then |\mathbf{a}+\mathbf{b}|^2 isa7b25c49d5
6. The angle between \mathbf{i}+\mathbf{j} and \mathbf{j}+\mathbf{k} isa60°b30°c45°d90°
7. The point dividing the join of the origin and 6\mathbf{i} in the ratio 1:2 has position vectora3\mathbf{i}b4\mathbf{i}c\mathbf{i}d2\mathbf{i}
8. Which of these is a unit vector?a\mathbf{i}+\mathbf{j}b\frac{\mathbf{i}+\mathbf{j}}{2}c\frac{\mathbf{i}+\mathbf{j}+\mathbf{k}}{\sqrt{3}}d\mathbf{i}+\mathbf{j}+\mathbf{k}