Question Video: Finding the Magnitude of Vectors | Nagwa Question Video: Finding the Magnitude of Vectors | Nagwa

# Question Video: Finding the Magnitude of Vectors Mathematics • Third Year of Secondary School

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If 𝐀 = ⟨1, −2, 2⟩, 𝐁 = ⟨2, 𝑚, 𝑛⟩, 𝐂 = ⟨𝑚, 𝑛, 𝑚 + 𝑛⟩, and 𝐀 ∥ 𝐁, find |𝐂|.

03:09

### Video Transcript

If vector 𝐀 is equal to one, negative two, two; vector 𝐁 is equal to two, 𝑚, 𝑛; vector 𝐂 is equal to 𝑚, 𝑛, 𝑚 plus 𝑛; and vector 𝐀 is parallel to vector 𝐁, find the magnitude of vector 𝐂.

We will begin by calculating the values of 𝑚 and 𝑛 using the fact that vectors 𝐀 and 𝐁 are parallel. We recall that if two vectors 𝐮 and 𝐯 are parallel, then 𝐯 is equal to 𝑘 multiplied by 𝐮, where 𝑘 is a scalar constant. In this question, vector 𝐁 is therefore equal to 𝑘 multiplied by vector 𝐀. The vector two, 𝑚, 𝑛 is equal to 𝑘 multiplied by one, negative two, two.

We can multiply a vector by a scalar by multiplying each of the components of the vector by that scalar. This means that the right-hand side of our equation becomes 𝑘, negative two 𝑘, two 𝑘. If any two vectors are equal, their corresponding components are equal. This means that two is equal to 𝑘, 𝑚 is equal to negative two 𝑘, and 𝑛 is equal to two 𝑘. Substituting 𝑘 equals two, we see that 𝑚 is equal to negative four. 𝑛 is equal to two multiplied by two, which is equal to positive four. This means that vector 𝐁 is equal to two, negative four, four. As vector 𝐂 was equal to 𝑚, 𝑛, 𝑚 plus 𝑛, this is equal to negative four, four, zero.

We are asked to find the magnitude of this vector. We can calculate the magnitude of any vector by finding the square root of the sum of the squares of its individual components. The magnitude of vector 𝐂 is equal to the square root of negative four squared plus four squared plus zero squared. Negative four squared and four squared are both equal to 16, and zero squared is zero. Our expression simplifies to the square root of 32.

Using our laws of radicals or surds, this can be rewritten as the square root of 16 multiplied by the square root of two. And as the square root of 16 is four, this is equal to four root two. If vectors 𝐀 and 𝐁 are parallel, then the magnitude of vector 𝐂 is four root two.

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