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Vectors - Intersecting Diagonals

Neil Trivedi

Teacher

Neil Trivedi

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Vectors - Intersecting Diagonals

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Challenging Question

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Vectors - Intersecting Diagonals

A vector is a quantity with both magnitude and direction, represented as a directed line or in coordinate form. Vectors are useful for studying diagonals in shapes such as parallelograms. They help find intersection points and check if diagonals bisect each other or divide in a specific ratio.

Example 1:

is a parallelogram, where  and . The diagonals and intersects at a point . Prove that the diagonals bisect each other.

A parallelogram 𝑂𝐴𝐵𝐶 is shown with vectors 𝑎 along 𝑂𝐴 and 𝑏 along 𝑂𝐵, a diagonal from 𝑂 to 𝐶 passing through point 𝑋, and an internal construction highlighting a smaller parallelogram near 𝑂𝐵.

Step 1: Let  and . and represent fractions between and that we are trying to find.

Note: We are first saying that is some fraction of and that is some fraction of . We need to show that both of those fractions (represented by and ) are .

Since  , we can write


Since , we can write



Step 2: Find  using two different paths, with one being direct and one being indirect.

A parallelogram 𝑂𝐴𝐵𝐶 is shown with vectors 𝑎 along 𝑂𝐴 and 𝑏 along 𝑂𝐵, a point 𝑄 on 𝑂𝐴, and the lines 𝐵𝑄 and 𝑂𝐶 intersecting at 𝑋, with an internal highlighted construction used to form vector equations.

Direct: 

Indirect:

Then collecting like terms by factorising from the first and last terms we are left with:


Step 3: Since is unique, the two vectors we found in step 2 must be equal. So, we equate the coefficients of and and solve.

       


Thus,  and . Therefore, the diagonals bisect each other.

No answer provided.

Example 2:

Given that , find the ratio of .

A parallelogram 𝑂𝐴𝐵𝐶 is shown with vectors 𝑎 along 𝑂𝐴 and 𝑏 along 𝑂𝐵, a point 𝑄 on 𝑂𝐴, and lines 𝐵𝑄 and 𝑂𝐶 intersecting at 𝑋 to illustrate a vector intersection problem.

Step 1: Let and . and  represent fractions between and that we are trying to find.

Since , we can write



Since , we can write



Step 2: Find using two different paths, with one being direct and one being indirect.

A parallelogram 𝑂𝐴𝐵𝐶 is shown with vectors 𝑎 along 𝑂𝐴 and 𝑏 along 𝑂𝐵, a point 𝑄 on 𝑂𝐴, and the lines 𝐵𝑄 and 𝑂𝐶 intersecting at 𝑋, with an internal highlighted construction used to form vector equations.

Direct:

Indirect:

Step 3: Since is unique, the two vectors we found in step 2 must be equal. So, we equate the coefficients of and and solve.


We now see that, .

Thus, the ratio is .

Note: If is of the diagonal line then the other part must be of the line.

Hence, the ratio .

No answer provided.

Challenging Question

Practice Questions

The Binomial Expansion