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Welcome to GCSE Edexcel Maths revision.

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Topic M 35: Vectors.

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This video covers Higher tier.

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It includes the shared content and the labelled Higher extensions.

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A vector has magnitude and direction.

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A column vector, 4, minus 3 represents 4 units right and 3 down.

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Its components are signed displacements, not a coordinate pair identifying one fixed point.

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Equal vectors have equal components even when drawn in different positions.

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The negative vector reverses direction: minus column vector, 4, minus 3 equals column vector, minus 4, 3 .

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Add component by component: column vector, 2, 3 plus column vector, 4, minus 1 equals column vector, 6, 2 .

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Geometrically, place the second arrow's tail at the first arrow's head.

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Head-to-tail vector addition

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Subtract using the reverse vector: column vector, 5, 2 minus column vector, 1, 4 equals column vector, 4, minus 2 .

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Order matters.

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Multiplying by a scalar changes length and may reverse direction. 3 column vector,

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2,

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minus 1 equals column vector,

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6,

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minus 3 ,

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while minus 2 column vector,

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2,

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minus 1 equals column vector,

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minus 4,

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2 .

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If A equals open bracket 1, 2 close bracket and B equals open bracket 5, minus 1 close bracket , vector A B is B minus A equals column vector, 4, minus 3 .

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Vector B A is its negative.

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A vector's magnitude follows Pythagoras: the magnitude of open bracket column vector,

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3,

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4 close bracket equals square root of open bracket 3 squared plus 4 squared close bracket equals 5.

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A translation's vector is measured in the coordinate system's units.

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Use bold letters such as a or directed labels such as vector A B for vectors.

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If O A equals a and O B equals b, then A B equals b minus a.

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The midpoint M of A B has O M equals 1 over 2 multiplied by open bracket a plus b close bracket .

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If M divides A B in ratio 1 to 2 from A, O M equals a plus 1 over 3 multiplied by open bracket b minus a close bracket .

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Vectors that are non-zero scalar multiples are parallel.

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To prove three points are collinear, show vectors along two joining segments are scalar multiples and share a point.

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Worked example: In triangle O A B, midpoints M of O A and N of O B have O M equals 1 over 2 a and O N equals 1 over 2 b.

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Therefore M N equals 1 over 2 multiplied by open bracket b minus a close bracket equals 1 over 2 A B, proving M N is parallel to A B and half its length.

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For a parallelogram OABC in order, if O A equals a and O C equals b, then O B equals a plus b.

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Diagonals can be expressed in more than one route to locate their intersection.

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Direction matters in each route: A B plus B C equals A C, but A B plus C B is different.

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Draw arrows and keep start and end labels consistent.

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For an intersection problem, express its position along each line with parameters, equate coefficients of independent vectors and solve the resulting simultaneous equations.

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That completes Vectors.

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Revisit the notes and test yourself on the revision website.