Concepts: Components, direction cosines, section formula and the dot product

Vector Algebra24 min readLesson 3 of 7

3.1 Component form Three unit vectors point along the coordinate axes: i along x, j along y and k along z. Any vector in space is a combination of these three. A point P with coordinates (x, y, z) has position vector OP = x i + y j + z k, and x, y, z are called the scalar components while x i, y j and z k are the vector components. The same vector can be written in a column or as a list, and both mean the same three numbers. Component form turns vector problems into arithmetic. The length is the square root of the sum of squares. Addition and subtraction work component by component. Multiplying by a scalar multiplies each component. Two vectors are equal exactly when all three corresponding components are equal, which is why one vector equation gives three scalar equations. For vectors a = a1 i + a2 j + a3 k and b = b1 i + b2 j + b3 k, the sum, the difference and a scalar multiple are:

a \pm b = (a_{1} \pm b_{1}) i^+ (a_{2} \pm b_{2}) j^+ (a_{3} \pm b_{3}) k^

λ a = (λ a_{1}) i^+ (λ a_{2}) j^+ (λ a_{3}) k^

a = b ⟺ a_{1} = b_{1}, a_{2} = b_{2}, a_{3} = b_{3}

3.2 Direction cosines and direction ratios Let a vector make angles alpha, beta and gamma with the positive directions of the x, y and z axes. The cosines of these angles are the direction cosines, usually written l, m and n. They are the components of the unit vector in that direction, so they satisfy a fixed identity: l squared plus m squared plus n squared equals one. Any three numbers proportional to the direction cosines are direction ratios; the components of the vector are the simplest choice of direction ratios. A single formula connects the two ideas. If a line has direction ratios a, b, c, then its direction cosines are each ratio divided by the square root of a squared plus b squared plus c squared, taken with one common sign. The common sign matters because a line can be traversed in two directions, and reversing it changes the signs of all three cosines.

l = \frac{x}{∣ r ∣}, m = \frac{y}{∣ r ∣}, n = \frac{z}{∣ r ∣}

l^{2} + m^{2} + n^{2} = 1

3.3 The vector joining two points and the section formula For points A(x1, y1, z1) and B(x2, y2, z2), the vector from A to B is found by subtracting the position vector of A from that of B. Its components are the differences of coordinates, and its length is the distance AB.

A B = (x_{2} - x_{1}) i^+ (y_{2} - y_{1}) j^+ (z_{2} - z_{1}) k^

A point R on the line AB that divides the segment in a ratio can be located by a formula in the position vectors of A and B. For internal division the parts stand on the same side, so the formula adds

r = \frac{m b + n a}{m + n} (internal), r = \frac{m b - n a}{m - n} (external)

Setting m equal to n in the internal formula gives the midpoint, which is the average of the two position vectors. This is a quick way to find the centre of a segment or to prove that one point is a m

3.4 The scalar product The dot product of two vectors measures how much they point along each other. It is defined as the product of their lengths and the cosine of the angle between them, and in component form it is the sum of the products of corresponding components. The result is a scalar, not a vector.

a \cdot b = ∣ a ∣∣ b ∣ cos θ = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}

Three consequences are used constantly. If the dot product is zero, nonzero vectors are perpendicular, since the cosine is zero at ninety degrees. If the vectors point the same way the product is the

The same formula rearranges into the angle between two vectors and into the projection of one vector on another. The projection is a signed length: it is positive for an acute angle, zero when the vectors are perpendicular and negative for an obtuse angle. That sign is information, not a mistake.

cos θ = \frac{a \cdot b}{∣ a ∣∣ b ∣}

projection of a on b = \frac{a \cdot b}{∣ b ∣}

Expanding a squared length is another standard move. The square of the length of a sum or difference contains the dot product in the middle term, which is how questions that give a length or a relatio

∣ a \pm b ∣^{2} = ∣ a ∣^{2} + ∣ b ∣^{2} \pm 2 a \cdot b

3.5 Worked examples Example 1. Take a = (2, 3, 6). Find its magnitude and its unit vector. The magnitude is the square root of 4 + 9 + 36, which is 7, so dividing each component by 7 gives the unit vector (2/7, 3/7, 6/7). Example 2. Find the direction cosines of the vector from P(1, 2, 3) to Q(4, 5, 6). The components of PQ are (3, 3, 3) and its length is 3 root 3, so each direction cosine is 1 divided by root 3, all with one sign. Example 3. Find the angle between a = i - j + k and b = i + j - k. The dot product is 1 - 1 - 1 = -1, and each length is root 3, so the cosine is minus one third. The angle is obtuse, which is why the negative sign is kept. Example 4. Find the projection of a = 2 i + 3 j + 2 k on b = i + 2 j + k. The dot product is 2 + 6 + 2 = 10 and the length of b is root 6, so the projection is 10 divided by root 6. Example 5. Find the position vector of the point dividing the join of A with position vector 2 i - j and B with position vector i + 2 j internally in the ratio 2 : 1. The formula gives (2 b + 1 a) divided by 3, which simplifies to 4 i + 3 j all over 3. Example 6. Three points have position vectors i - 2 j + 3 k, 2 i + j and 3 i + 4 j - 3 k. Prove that they lie on one line. Taking differences, the first two vectors differ by i + 3 j - 3 k and the next pair by i + 3 j - 3 k again. The two joining vectors are equal, so the three points lie on one line.

Exam

When a question gives a relation between lengths, square it and expand. When it gives a projection, use the dot product divided by the magnitude of the base vector. When it gives a ratio on a line, write the section formula with the ratio in the same order as the points named. Each of these turns a word problem into two lines of arithmetic.

Common mistake

Do not confuse direction ratios with direction cosines, and do not forget to divide by the magnitude. Do not drop the sign of a projection or of a cosine: an obtuse angle gives a negative value. And in the section formula, check whether the division is internal or external before substituting.