Video Transcript
What is the property that shows that 𝑐 multiplied by vector 𝐚 plus vector 𝐛 is equal to 𝑐 multiplied by vector 𝐚 plus 𝑐 multiplied by vector 𝐛?
On the left-hand side of our equation, we are multiplying the scalar 𝑐 by the sum of two vectors 𝐚 and 𝐛. Since this is equal to 𝑐 multiplied by vector 𝐚 plus 𝑐 multiplied by vector 𝐛, this is an example of the distributive property of scalar multiplication over vector addition. We distribute the scalar across the two vectors. Recalling the distributive property of multiplication, we can multiply 𝑥 plus 𝑦 by two by distributing the two across the parentheses. This gives us two 𝑥 plus two 𝑦.
If we let vector 𝐚 have components 𝑎 sub 𝑥 and 𝑎 sub 𝑦 and vector 𝐛 have components 𝑏 sub 𝑥 and 𝑏 sub 𝑦, we can see how this also works for vectors. Multiplying the sum of these vectors by 𝑐, we have 𝑐 multiplied by 𝑎 sub 𝑥, 𝑎 sub 𝑦 plus 𝑏 sub 𝑥, 𝑏 sub 𝑦. We know that to add two vectors, we add their corresponding components. This means that we have 𝑐 multiplied by 𝑎 sub 𝑥 plus 𝑏 sub 𝑥, 𝑎 sub 𝑦 plus 𝑏 sub 𝑦. Next, to multiply the vector by scalar 𝑐, we multiply each component by 𝑐, giving us 𝑐 multiplied by 𝑎 sub 𝑥 plus 𝑏 sub 𝑥, 𝑐 multiplied by 𝑎 sub 𝑦 plus 𝑏 sub 𝑦.
Next, we know that multiplication is distributive over addition, giving us the vector 𝑐𝑎 sub 𝑥 plus 𝑐𝑏 sub 𝑥, 𝑐𝑎 sub 𝑦 plus 𝑐𝑏 sub 𝑦. This can be rewritten as the vector 𝑐𝑎 sub 𝑥, 𝑐𝑎 sub 𝑦 plus the vector 𝑐𝑏 sub 𝑥, 𝑐𝑏 sub 𝑦. Factoring out the scalar 𝑐 from both of our vectors, we have 𝑐 multiplied by 𝑎 sub 𝑥, 𝑎 sub 𝑦 plus 𝑐 multiplied by 𝑏 sub 𝑥, 𝑏 sub 𝑦, which is equivalent to 𝑐 multiplied by vector 𝐚 plus 𝑐 multiplied by vector 𝐛. This is the distributive property of scalar multiplication over vector addition.
