Vector Basics (AA HL)
Before lines, planes, or the cross product, every vector question rests on a handful of simple moves: adding and subtracting components, scaling a vector, and locating a point along a line segment. Get these fluent and the harder HL vector work stops being a source of careless errors. It's part of the broader Vectors, Lines & Planes topic.
8 questions on this sub-topic.
Key ideas and formulas
Covered under IB syllabus reference AHL3.12: the concept of a vector, position and displacement vectors, representation using directed line segments, and the base vectors \(\mathbf i,\mathbf j,\mathbf k\).
Magnitude of a vector
\(|\mathbf v|=\sqrt{x^2+y^2+z^2}\)
Not in the formula booklet - it's an extension of Pythagoras' theorem into three dimensions, and you're expected to know it from prior learning.
Point dividing a line segment
\(\overrightarrow{OP}=\mathbf a+t(\mathbf b-\mathbf a)\)
Locates a point \(P\) that splits \(AB\) in a given ratio: write the ratio as a fraction \(t\) of the way from \(A\) to \(B\), then substitute.
Ready for the dot product, cross product, or full syllabus wording? See Vectors, Lines & Planes.
Worked examples
Given \(\mathbf a=(1,3,-2)\) and \(\mathbf b=(4,-1,5),\) find \(\mathbf a+\mathbf b.\)
Worked solution
\(\mathbf a + \mathbf b = (1+4, 3-1, -2+5)\) M1
\(= (5, 2, 3).\) A1
Find the point \(P\) dividing \(AB\) in the ratio \(2:1\), where \(A(1,2,3),\ B(7,8,9).\)
Worked solution
\(P = \mathbf a + \tfrac23(\mathbf b - \mathbf a).\) M1
\(\mathbf b - \mathbf a = (6,6,6).\) A1
\((1,2,3) + \tfrac23(6,6,6) = (1+4, 2+4, 3+4)\) M1
\(= (5,6,7).\) A1
Given \(\mathbf a=(1,-2,3)\) and \(\mathbf b=(4,0,-1)\), find \(2\mathbf a-\mathbf b.\)
Worked solution
\(2\mathbf a = (2, -4, 6).\) M1
\(2\mathbf a - \mathbf b = (2-4, -4-0, 6+1)\) A1
\(= (-2, -4, 7).\) A1
Find \(p\) so that \(\mathbf a = (4, p, 6)\) is parallel to \(\mathbf b = (2, 3, 3).\)
Worked solution
Parallel means \(\mathbf a = k\mathbf b.\) From the first component \(4 = 2k \Rightarrow k\) M1
\(= 2.\) A1
Then \(p = 3k = 6.\) A1
Common mistakes
- Writing the cross product as a number. Even at the "basics" stage it's worth fixing this early: the vector product always produces a vector - give the answer as \((x,y,z)\), never as a single scalar.
- Using the wrong fraction for a ratio. A ratio of \(2:1\) means \(P\) is \(\tfrac23\) of the way from \(A\) to \(B\), not \(\tfrac21\) or \(\tfrac12\) - divide the first part of the ratio by the sum of both parts.
- Mixing up position and displacement vectors. \(\overrightarrow{OA}\) is the position vector of \(A\) from the origin, while \(\overrightarrow{AB}=\mathbf b-\mathbf a\) is a displacement between two points - confusing the two leads to the wrong subtraction order.
Ready to practise properly?
11 vector-basics questions, marked instantly like the real exam.
Quick answers
How do you add or subtract vectors?
Add or subtract componentwise: combine the \(x\)-components together, the \(y\)-components together, and the \(z\)-components together.
How do you find a point that divides a line segment in a given ratio?
Convert the ratio to a fraction \(t\) of the way from the first point to the second, then use \(P = \mathbf a + t(\mathbf b - \mathbf a)\).