   Chapter 12.2, Problem 50E

Chapter
Section
Textbook Problem

# Prove Property 5 of vectors algebraically for the case n =3. Then use similar triangles to give a geometric proof.

To determine

To prove: Algebraic and geometrical proof of property 5 of vectors.

Explanation

Given:

The value of n is 3.

Formula:

Consider the two vectors as a=a1,a2,a3 and b=b1,b2,b3.

Sum of vectors:

The vector sum of two vectors (a+b) is,

a+b=a1,a2,a3+b1,b2,b3=a1+b1,a2+b2,a3+b3

Scalar multiplication with a vector:

The scalar multiplication of vector (ca) is,

ca=ca1,a2,a2=ca1,ca2,ca3

Here,

c is scalar.

Triangle law:

Consider two vectors u and then vector v. Draw the vector u and second vector v at the end of vector u, then the resultant vector would be the sum of two vectors (u+v).

Write the expression for property 5.

c(a+b)=ca+cb (1)

Here,

A and b are vectors, and

c is scalar.

The value of n is 3, hence vectors are three-dimensional. Consider the vectors as a=a1,a2,a3 and b=b1,b2,b3.

Substitute a1,a2,a3 for a and b1,b2,b3 for b in equation (1),

c(a1,a2,a3+b1,b2,b3)=ca1,a2,a3+cb1,b2,b3ca1+b1,a2+b2,a3+b3=ca1,ca2,ca3+cb1,cb2,cb3

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