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Animated Solution for Mathematics - Vector Algebra: If the vectors $A B = 3 \hat{i} + 4 \hat{k}$ and $A C = 5 \hat{i} - 2 \hat{j} + 4 \hat{k}$ are the sides of a triangle $A B C$ , then the length of the median through $A$ is

Q: Wait, why is the median vector the average of the two side vectors?

Think of the parallelogram formed by $\vec{AB}$ and $\vec{AC}$. The diagonal of this parallelogram is $\vec{AB} + \vec{AC}$. The median $AM$ is exactly half of this diagonal. It is a fundamental property of vector geometry that the median from a vertex is the average of the two adjacent side vectors.

Select Answer:

Visualized Solution

Visualizing the Triangle $A B C$

Given vectors representing sides of $△ A B C$ :
$A B = 3 \hat{i} + 4 \hat{k}$
$A C = 5 \hat{i} - 2 \hat{j} + 4 \hat{k}$

The Median Vector Formula

Let $M$ be the midpoint of side $B C$ .
The median vector $A M$ from vertex $A$ is given by:
$A M = \frac{A B + A C}{2}$

Setting up the Vector Addition

Substitute the given vectors into the formula:
$A M = \frac{( 3 i ^ + 0 j ^ + 4 k ^ ) + ( 5 i ^ - 2 j ^ + 4 k ^ )}{2}$

Adding the $\hat{i}$ Components

Grouping the $\hat{i}$ terms:
$(3 + 5) \hat{i} = 8 \hat{i}$

Adding the $\hat{j}$ Components

Grouping the $\hat{j}$ terms:
$(0 - 2) \hat{j} = - 2 \hat{j}$

Adding the $\hat{k}$ Components

Grouping the $\hat{k}$ terms:
$(4 + 4) \hat{k} = 8 \hat{k}$

Calculating the Median Vector $A M$

Divide the sum by $2$ :
$A M = \frac{8 i ^ - 2 j ^ + 8 k ^}{2}$
$A M = 4 \hat{i} - \hat{j} + 4 \hat{k}$

Formula for Magnitude

The length of a vector $V = x \hat{i} + y \hat{j} + z \hat{k}$ is its magnitude:
$∣ V ∣ = x^{2} + y^{2} + z^{2}$

Substituting Components for Length

Substitute the components of $A M$ into the magnitude formula:
$∣ A M ∣ = (4)^{2} + (- 1)^{2} + (4)^{2}$

Squaring the Components

Calculate the squares:
$4^{2} = 16$
$(- 1)^{2} = 1$
$∣ A M ∣ = 16 + 1 + 16$

Final Calculation

Sum the values inside the square root:
$16 + 1 + 16 = 33$
$∣ A M ∣ = 33$

Conclusion

Final Answer:
The length of the median through $A$ is $33$ .

00:00 / 00:00

The Sigma Insight: Addition of Vectors

Solution Diagram

Analyzing the Setup

Imagine you are standing at vertex

A

of a triangle

A B C

. You are given two vectors,

A B = 3 \hat{i} + 4 \hat{k}

and

A C = 5 \hat{i} - 2 \hat{j} + 4 \hat{k}

, which define the sides of your triangle.

These vectors represent paths leading away from your current position. The problem requires finding the length of the median through

A

, which is the line segment connecting

A

to the midpoint

M

of the opposite side

B C

The Master Formula

To find the median vector

A M

, we utilize the parallelogram law. If we complete the parallelogram with sides

A B

and

A C

, the diagonal starting from

A

is the sum

A B + A C

The median

A M

is exactly half of this diagonal. Thus, our master formula is:

A M = \frac{A B + A C}{2}

The Calculation

Now, we perform the vector addition. We express the vectors as

A B = 3 \hat{i} + 0 \hat{j} + 4 \hat{k}

and

A C = 5 \hat{i} - 2 \hat{j} + 4 \hat{k}

. Adding these component-wise, we obtain:

A M = \frac{( 3 + 5 ) i ^ + ( 0 - 2 ) j ^ + ( 4 + 4 ) k ^}{2}

This simplifies to:

A M = \frac{8 i ^ - 2 j ^ + 8 k ^}{2}

Dividing each component by

2

, we arrive at the median vector:

A M = 4 \hat{i} - 1 \hat{j} + 4 \hat{k}

The Final Stretch

Magnitude

The question asks for the length of this median, which is the magnitude of the vector

A M

. The formula for the magnitude of a vector

V = x \hat{i} + y \hat{j} + z \hat{k}

∣ V ∣ = x^{2} + y^{2} + z^{2}

Substituting our components, we get:

∣ A M ∣ = 4^{2} + (- 1)^{2} + 4^{2}

Squaring these values yields

16 + 1 + 16

, which sums to

33

. Therefore, the length of the median is $33$ .

Common Traps & FAQs

Why can't we just find the coordinates of

B

and

C

and use the distance formula?

You absolutely could, but it is a much longer path. You would need to find the coordinates of

B

and

C

relative to

A

, then find the midpoint

M

, and finally calculate the distance

A M

. Vector addition is a direct, elegant shortcut that avoids unnecessary coordinate geometry.

Wait, why is the median vector the average of the two side vectors?

Think of the parallelogram formed by

A B

and

A C

. The diagonal of this parallelogram is

A B + A C

. The median

A M

is exactly half of this diagonal. It is a fundamental property of vector geometry that the median from a vertex is the average of the two adjacent side vectors.

Does the order of subtraction matter when calculating the magnitude?

Not at all! Because the magnitude formula involves squaring each component (

x^{2} + y^{2} + z^{2}

), any negative signs are eliminated. Whether you have

- 1

1

, the square is always

1

. Just be careful with your arithmetic!

Animated Solution for Mathematics - Vector Algebra: If the vectors $A B = 3 \hat{i} + 4 \hat{k}$ and $A C = 5 \hat{i} - 2 \hat{j} + 4 \hat{k}$ are the sides of a triangle $A B C$ , then the length of the median through $A$ is

Select Answer:

Visualized Solution

Visualizing the Triangle $A B C$

The Median Vector Formula

Setting up the Vector Addition

Adding the $\hat{i}$ Components

Adding the $\hat{j}$ Components

Adding the $\hat{k}$ Components

Calculating the Median Vector $A M$

Formula for Magnitude

Substituting Components for Length

Squaring the Components

Final Calculation

Conclusion

The Sigma Insight: Addition of Vectors

Analyzing the Setup

The Master Formula

The Calculation

The Final Stretch

Similar Questions

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A vector $a = α \hat{i} + 2 \hat{j} + β \hat{k} (α, β \in R)$ lies in the plane of the vectors, $b = \hat{i} + \hat{j}$ and $c = \hat{i} - \hat{j} + 4 \hat{k}$ . If $a$ bisects the angle between $b$ and $c$ , then:

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Select Answer:

Visualized Solution

Visualizing the Triangle .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ABC

The Median Vector Formula

Setting up the Vector Addition

Adding the .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } i^ Components

Adding the .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } j^​ Components

Adding the .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } k^ Components

Calculating the Median Vector .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } AM

Formula for Magnitude

Substituting Components for Length

Squaring the Components

Final Calculation

Conclusion

The Sigma Insight: Addition of Vectors

Analyzing the Setup

The Master Formula

The Calculation

The Final Stretch

Similar Questions

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Adding the $\hat{i}$ Components

Adding the $\hat{j}$ Components

Adding the $\hat{k}$ Components

Calculating the Median Vector $A M$

The vectors $A B = 3 \hat{i} + 4 \hat{k}$ and $A C = 5 \hat{i} - 2 \hat{j} + 4 \hat{k}$ are the sides of a triangle ABC. The length of the median through A is

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