If the two vectors 𝐀 which is equal to five, negative 10 and 𝐁 which is equal to two, 𝑘 are perpendicular, determine the value of 𝑘.
We recall that if two vectors 𝐮 and 𝐯 are perpendicular, then their dot or scalar product is equal to zero. We can calculate the dot product as follows. If vector 𝐮 has components 𝑢 sub one and 𝑢 sub two and vector 𝐯 has components 𝑣 sub one and 𝑣 sub two, then the dot product of 𝐮 and 𝐯 is equal to 𝑢 sub one multiplied by 𝑣 sub one plus 𝑢 sub two multiplied by 𝑣 sub two.
In this question, we need to calculate the dot product of vector 𝐀 and vector 𝐁. This is equal to five multiplied by two plus negative 10 multiplied by 𝑘. This in turn simplifies to 10 minus 10𝑘. As the vectors are perpendicular, this is equal to zero. We can then add 10𝑘 to both sides of our equation such that 10𝑘 is equal to 10. Finally, dividing both sides of this equation by 10 gives us 𝑘 is equal to one. If the two vectors five, negative 10 and two, 𝑘 are perpendicular, then 𝑘 is equal to one.