If vector v ⃗ = ⟨ − 4 , − 10 ⟩ v ⃗=⟨-4,-10⟩ v ⃗ = ⟨ − 4 , − 10 ⟩ , calculate the magnitude ∣ v ⃗ ∣ |v ⃗ | ∣ v ⃗∣ .

Let's try solving this problem. So in this problem, we are given the vector negative 4 comma negative 10, and we're asked to calculate the magnitude of this vector. Now the equation for calculating the magnitude of any vector is to take the x component and square it, and add it to the y component squared, and then you wanna take that whole thing and put it underneath the square root. This is basically just the pythagorean theorem, except we're doing this with the components of the vector. Now with the vector we have up here, I can see that the x component we have is negative 4. So we're going to have negative 4 squared plus the y component squared, which is negative ten squared and then that's all gonna be underneath the square root. Now negative four squared comes out to positive 16 and negative ten squared comes out to positive a 100. The reason that we got positive numbers is because when you you square a negative number, the negative sign cancels itself. Because negative four squared is the same thing as negative four times negative four. And the negative signs would cancel, so you just have 4 times 4, which is 16. And the same thing with this negative 10, you get positive a 100. So we have the square root of 16 plus a 100, which is the square root of a 116. Now this here is the answer to the problem, but it turns out we actually can simplify that what's underneath this square root. So let's go ahead and do that to make sure we get the best answer we can or the most simplified. So to do this, the square root of a 116, that turns out to be the same thing as the square root of 4 times 29. You can check this. 4 times 29 is equal to 116. And 4 is the same thing as 2 squared. So we could write this as 2 squared times 29. And what we can do from here is split this into 2 square roots, so we can have the square root of 2 squared times the square root of 29. Now the square and the square root will cancel each other, meaning we'll just have 2, and then that's all going to be multiplied by the square root of 29. So for our vector v in its most simplified form, we're going to have 2 times the square root of 29, and that is the magnitude of this vector. So hope you found this video helpful, thanks for watching.

If vector v ⃗ = ⟨ − 4 , − 10 ⟩ v ⃗=⟨-4,-10⟩ v ⃗ = ⟨ − 4 , − 10 ⟩ , calculate the magnitude ∣ v ⃗ ∣ |v ⃗ | ∣ v ⃗∣ .

√ − 116 \surd-116 √ − 116

14. Vectors

Vectors in Component Form

Precalculus

14. Vectors

Vectors in Component Form

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Multiple Choice

If vector $v ⃗=⟨-4,-10⟩$ , calculate the magnitude $|v ⃗ |$ .

$4\surd29$

$2\surd116$

$2√29$

$\surd-116$

Verified step by step guidance

Understand that the magnitude of a vector v⃗ = ⟨a, b⟩ is calculated using the formula ∣v⃗∣ = √(a² + b²). This formula is derived from the Pythagorean theorem, which applies to the components of the vector.

Identify the components of the vector v⃗ = ⟨−4, −10⟩. Here, a = −4 and b = −10.

Substitute the values of a and b into the magnitude formula: ∣v⃗∣ = √((-4)² + (-10)²).

Calculate the squares of the components: (-4)² = 16 and (-10)² = 100.

Add the squared values and simplify the expression: ∣v⃗∣ = √(16 + 100) = √116. Then, simplify √116 to its simplest radical form.

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Struggling with Precalculus?

If vector v⃗=⟨−4,−10⟩v ⃗=⟨-4,-10⟩v⃗=⟨−4,−10⟩, calculate the magnitude ∣v⃗∣|v ⃗ |∣v⃗∣.

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Vectors in Component Form practice set

If vector $v ⃗=⟨-4,-10⟩$ , calculate the magnitude $|v ⃗ |$ .