Question

Can the sum of the magnitudes of two vectors ever be equal to the magnitudes of the some of two vectors?

Answer 1

Explanation:

IT IS ≥ OR EQUAL TO.

THANKS AND MARK ME AS BRAINLIEST.

Answer 2

Yes, the sum of magnitude of two vectors can be equal to the magnitude of sum of these two vectors.

Consider two vectors\overrightarrow{a} = a_{x}\widehat{i} +a _{y}\widehat{y} and such that the sum of both the vectors is given by another vector as:

\overrightarrow{c} = \overrightarrow{a} + \overrightarrow{b}

Substitute\overrightarrow{a} = a_{x}\widehat{i} + a_{y}\widehat{j} and , \overrightarrow{b} = b_{x}\widehat{i} + b_{y}\widehat{j}

\overrightarrow{c} = a_{x}\widehat{i} + a_{y}\widehat{j} + b_{x}\widehat{i} + b_{y}\widehat{j}

= (a_{x} + b_{x})\widehat{i} + (a_{y} + b_{y})\widehat{j}

The magnitude of vector \overrightarrow{c}is:

|\overrightarrow{c}| = \sqrt{(a_{x}+b_{x})^{2} + (a_{y}+b_{y})^{2}}

The magnitude of vectors is: \overrightarrow{a} = a_{x}\widehat{i} + a_{y}\widehat{j}

|\overrightarrow{a}| = \sqrt{(a_{x})^{2}+ (a_{y})^{2}}

The magnitude of vectors\overrightarrow{b} = b_{x}\widehat{i} + b_{y}\widehat{j} is:

|\overrightarrow{b}| = \sqrt{(b_{x})^{2}+ (b_{y})^{2}}

If the sum of magnitude of vector\overrightarrow{a} and magnitude of vector\overrightarrow{b} is equal to magnitude of vector\overrightarrow{c} , then

|\overrightarrow{c}| = |\overrightarrow{b}| + |\overrightarrow{a}|

Square both sides and expand,

|\overrightarrow{c}|^{2} = \left ( |\overrightarrow{b}| \right + |\overrightarrow{a}|)^{2}

= |\overrightarrow{a}|^{2} + |\overrightarrow{b}|^{2} + 2 |\overrightarrow{a}| |\overrightarrow{b}|

Substitute|\overrightarrow{c}| = \sqrt{(a_{x}+ b_{x})^{2} + (a_{y}+ b_{y})^{2}} , |\overrightarrow{a}| = \sqrt{(a_{x})^{2} + (a_{y})^{2}}and |\overrightarrow{b}| = \sqrt{(b_{x})^{2} + (b_{x})^{2}},

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Square both sides,

Therefore, if two vectors are such that the condition2a_{x}a_{y}b_{x}v_{y} = a_{x}^{2}b_{y}^{2}+ a_{y}^{2}b_{x}^{2} is satisfied, then the magnitude of the sum of two vectors is equal to the sum of magnitude of the vectors.

For example, assume \overrightarrow{b} = -\widehat{i}-\widehat{j}and\overrightarrow{b} = -\widehat{i}-\widehat{j} , then the condition 2a_{x}a_{y}b_{x}b_{y} = a_{x}^{2} b_{y}^{2}+ a_{y}^{2} b_{x}^{2} gives 2 = 2 , therefore for these vectors that sum of magnitude of each vector is equal to the magnitude of the sum of the vectors